Sophie Dahl wants a cigarette. She bounds into her publisher's offices, a tumbling blur of gazelle legs, bushbaby eyes and leopard-print shawl, asking if there might just be time for a quick hit of nicotine. I'm mid-sentence, joking with Dahl's publicist about the ultimate in vicious opening questions: what's your opinion of Nietzsche? "I really wouldn't have wanted to talk about Nietzsche," Dahl breaks in. She pronounces the name more confidently, and, I suspect, more accurately, than I did. Then she turns heel for a sneaky one outside.
The woman formerly known as "Britain's largest model" (a size 14 at her biggest - imagine!) is clearly no fool. We are here to talk about her first novel, Playing With the Grown-ups, a coming-of-age tale that Dahl, 30, has described as "somewhat autobiographical". To sum up the plot: adolescent protagonist Kitty grows up in a sprawling family, with an absent father, some half-siblings and a funny, flighty, depressive mother who carts them off across the globe in search of God.
Dahl has been published before - short stories in magazines and newspapers, and an illustrated novella, The Man with the Dancing Eyes. But where much of her earlier writing was highly whimsical (sometimes prompting a teetering fear that a unicorn might march into the action at any moment), this novel is slight but deftly written. Peopled by great British eccentrics, it is oddly ageless - aside from a few minor details, it could be set in the 1890s, 1920s or 1950s. It is no surprise that Nancy Mitford and Evelyn Waugh are among her favourite writers.
Returning from her cigarette, Dahl folds herself into the chair opposite me. Did she enjoy writing the book, I ask. Her eyes widen. "Yes, well, I loathed it in the beginning, totally loathed it." "Oh God," she thought, "I can't write a whole book with no pictures!" She emits her trademark laugh, a guffaw that sounds as though she is simultaneously exploding and imploding. "Because I didn't quite have the confidence, not quite trusting my own voice and the voice of the book, there were definitely some teething problems ..."
When it comes to writing, Dahl has a towering ancestry to contend with. Her maternal grandfather was Roald Dahl, to whom she was very close as a bespectacled, bookish, Enid Blyton-loving child. (He named the "poor little scrumplet" at the centre of the BFG after her.) Her mother, Tessa Dahl (who had Sophie at 19 during a short relationship with Carry On actor Julian Holloway), is also a writer. Does the thought of being measured against her grandfather bother her? She smiles. "Well, it's something that I've had to have a sense of humour about, because if I was really to take that bait, I wouldn't do a thing. That legacy is so huge, and I will never pretend to be capable of that ... I'm in as much awe of him as a writer as anyone else. It's not as though I'm thinking" - she adopts the voice of an evil prince -"Oh good, the crown is now mine!"
Fortunately, Roald Dahl was best known as a children's author, which is a little different from what she is doing. "Yes," she agrees, "and I think all those comparisons were probably more difficult for my mum, because she's only one generation away."
But that was the least of Tessa Dahl's problems. Sophie's mother had a horrible childhood - her older sister died from encephalitis, her baby brother was hit by a taxi and suffered brain damage, and her mother, the Oscar-winning actress Patricia Neal, suffered a series of debilitating strokes while giving Tessa a bath. Tessa went on to become a gossip page fixture (dating Peter Sellers and Brian de Palma, among others), and later spoke openly about her manic depression and drug addiction. In pursuit of happiness, she whisked the young Sophie through 10 schools and 17 homes in locations including London, New York, and an ashram in India. She has had three other children, two brief marriages, and once attempted suicide.
The shadows of Tessa Dahl clearly colour the mother figure in the novel - a woman who takes drugs, is devoted to a swami and self-harms - but what is really striking is just how warm the portrait is. Sophie Dahl has said she longed for stability as a child, yet she clearly also appreciates the sudden bolts of excitement and strangeness that shot through her early years. Does the love between the mother and daughter characters reflect her own experience? "I hope so, and I think if anything's autobiographical, it's that. I have a great deal of admiration for my mum. I think she's incredibly strong and she's coped with a great, great deal, and she's managed to have a huge sense of humour and compassion. She's made her depression public, so I'm not telling tales out of school, but she's had a very hard time of it." She adds that Tessa has been writing in earnest again. "She's just finished a novel, her second adult novel, and I'm so pleased, because she's a writer in every bit of her."
In these days of misery memoirs and therapy, of adults blaming their temper on that time when they were 10 and their parents denied them extra ice cream, Dahl seems refreshingly well-adjusted. She shrugs. "My childhood was such an odd one, but with such magic, and the quirky grown-ups who were in it managed to still bring a huge sense of love and magic, so for that I'm really grateful. Also, I think I really hate bitter books, and bitter people."
It was a row with her mother that first launched Dahl into the public eye. Having flunked out of school ("I didn't finish my A-levels because I didn't really go to school very much"), she had harboured hopes of heading to Florence to study art history. "I wanted to walk around a market with a basket bursting full of figs and cherries in a Sophia Loren dress," she says. "My parents' plan was, 'No, this will not be happening. You have not earned the boon of going to Florence. You will go and do a secretarial course, and you will go and get a job until you've actually proved your worth.' "
Dahl duly stropped off to secretarial college. But then, aged 18, she bumped into the stylist and fashion eccentric Isabella Blow. "She got out of a cab in a Philip Treacy hat and a McQueen corset, sat down next to me, and said 'Wow, you can be a model!' So we went off to find my mum, who was, of course, sooooo annoyed. I was so smug about the entire thing." The laughter peals out of her.
More than a decade on, the reaction to her first fashion shoots still seems astounding. She was called a "big friendly giant", and attracted captions such as "Sophie and the giant peaches". Commentators pondered whether hundreds of larger models would follow in her wake, and interviewers asked whether it was possible for her to find clothes that would fit her (she was, I repeat, just a size 14). Most of these articles were written by women journalists, who dressed up their scrutiny of Dahl's body as a celebration of her curves, excitement at the arrival of a new Volupta. A distinct current of criticism ran through much of the coverage, though, and there was no hiding the fact that this was just another instance of objectification, of a woman's body being claimed as public property and pored over with pornographic delight.
In early interviews Dahl was incredibly open, but over the years she seems to have become more restrained. "I started when I was 18," she points out, "and when people would ask me what I thought about something they'd get a slightly uncensored stream ..." she tails off. "I was puppyish, and I was really excited, because I was getting paid to wear lovely clothes and go to parties and get my hair and makeup done. I was probably the least jaded person - I wanted to be friends with everyone, so if anything's changed it's that I'm slightly more insular, for want of a better word. There are certainly still elements of that though - I'm a bit of a dork, really."
Dahl still models occasionally, and her body is still subject to intense scrutiny. When she decided to lose weight a few years back, writers wondered whether she was getting too thin - and, not only that, but if her weight loss was a betrayal of all womankind. Then, when she started going out with the jazz singer Jamie Cullum earlier this year, attention turned to her height. He is 5ft 6in; she is 6ft.
Dahl was shocked by the gawping she attracted when she started modelling - it seems strange that she wasn't fully forewarned by those around her, but, to be fair, given the extent of the coverage, even a ton of advice might not have prepared her properly. "It was certainly odd," she says. "My size wasn't something that I'd ever spent a huge amount of time thinking about - I guess at the age of 17 or 18 you don't. You know, I just wanted to be reasonably the same shape and size as my friends so that I could borrow clothes from them. It was as simple as that. So to suddenly have what should be a private time, when you're working out your relationship to your body, made very public was just extraordinary. It used to make me feel incredibly uncomfortable. Those photographs that should be fuel for tender family moments, or jokes between you and your best girlfriends, are suddenly all out there on the internet, for everyone to see for as long as they like."
Does the current obsession with female imperfection - the trend for magazines to circle a celebrity's scuffed fingernail or visible vein - surprise her? "I'm hoping that there will be a turnaround," she says. "I don't think that it can really sustain itself for that long."
When it comes to Cullum, Dahl is tightlipped, but clearly very happy. "We met at a charity thing that I sang at, and we became friends and it went from there. He accompanied me - I sang I Think it's Going to Rain Today, by Nina Simone - a tricky song to sing in front of 800 people." And do the comments about their height difference bother them? She says that she finds it weird, but that "when you're really happy you don't give a shit. I think if we were both slightly more neurotic people, it would be crippling, but the fact of the matter is that we're both happy. It's cool."
She loves writing, would like to spend the rest of her life doing it, and is currently working on a non-fiction project: a memoir of food. "Because I cook a lot, I wanted to write a recipe book, really incorporating the message that you don't have to starve yourself to be reasonably skinny. And that you can eat quite happily, and that the more one obsesses about it, the more of an issue it becomes."
Based in the US for much of the past decade, she is happily settled back in the UK, because, she says, "I got that terrible longing for home and the Antiques Roadshow and a good cup of tea. All those really mundane things." She is on the way to becoming a British institution, I say, immediately grimacing at my own words. Who wants to be called an institution at the age of 30? "Oh," she says. "Don't worry. I'd much prefer to be a great British institution than the great British body. Or rather," she rolls her eyes, "the undoing of the great British body".
·Playing with the Grown-ups is published by Bloomsbury, price £12.99. To order a copy for £11.99 with free UK p&p go totheguardian.com/bookshop or call 0870 836 0875.