When Deborah Levy was a schoolgirl in Johannesburg, she got into trouble with her teacher for ignoring the directions to always begin on the top line when writing in her exercise book. “I did not know why,” she remembers in Things I Don’t Want to Know, her 2013 meditation on writing, “but I always started on the third line so there was a gap between the top of the page and the line I started on; writing on the top line was impossible.”

Fifty years later, Levy is still leaving provocative gaps in her writing. Her latest novel, Hot Milk, a July release from Bloomsbury USA, doles out information in an elliptical, evocative fashion that will be familiar to readers of Swimming Home, which was shortlisted for the Booker in 2012. In Hot Milk, narrator Sofia Papastergiadis has brought her mother, Rose, to a clinic in southern Spain in hopes of learning the cause of Rose’s mysterious, intermittent difficulties in walking.

Emulating in her own distinctive style the great modernists she names as her primary literary influences (Beckett, Eliot, Pound, Woolf), Levy peels back narrative layers to reveal a troubled family history and a protagonist grappling with issues of identity and power that roil much of her fiction. “Sofia’s first problem is cultural identity,” Levy says. “She has a mother from northern Britain, and a Greek father who walked out when she was five. She has this name nobody can pronounce or spell, and she has to carry an identity she knows very little about. Also, I wanted to take a young woman who is full and quiet—not really sure where she’s heading, a little bit lost—and give her some room to make mistakes and experiment. It is so hard to say the things we want and the things we feel; all my writing circles that idea. I want to make a space for the difficulty of language, which we all experience every single day.”

Hot Milk is clearly much more than “a novel about hypochondria,” though Levy frequently described it that way when it was a work in progress. “I would describe it now as a thriller of symptoms,” she says, chuckling. She’s spent a lot of time talking and thinking about Hot Milk since it was published in the U.K. three months ago. Now, visiting New York from her home in London, she settles into a chair in the Vinyl Room of Manhattan’s members-only Soho House and continues: “I’m interested in the idea—Freud’s idea, but it goes back to the Greeks—that symptoms do the speaking for us, things that are too awkward to say. Hypochondriacs don’t really want a diagnosis; if a physician comes near with one, they might say no and present even more unfathomable symptoms. You know, when physicians take a case history they call it a narrative, and narrative is my subject as a writer; I’m obsessed with it. The idea that a physician is taking a narrative and that a hypochondriac might want to flee from that narrative really appealed to me.”

As a teenager in London, Levy herself needed to break out of a narrative constructed by others. Her family moved to England in 1968, when she was nine, after her father served a five-year jail stint because of his membership in the banned African National Congress. “The fight against racism and for democracy was the story of my parents’ generation,” she says. “But when I came to Britain I wanted to make another story for myself. When I was 15, I had read all about the bohemians in the South Bank in Paris, drinking their espressos, and I thought, that sounds great, I want to do that! So I used to write on napkins, it was just a rehearsal, an imitation of a writer, and it still makes me laugh to think about it now.”

Although her earliest writing was poetry (“I was in this group; we would read in pubs with shaking hands”), Levy trained as a playwright at the Dartington College of Arts and had a substantial career in the theater during the 1980s; her plays were produced everywhere from the Royal Shakespeare Company to London’s fringe (the equivalent of off-Broadway). “I worked very extensively and passionately in the theater,” she says. “It’s very communal. You go into the rehearsal room, and there are the actors, the director, the designers; you give over your play to them to interpret. But after a while I wanted to sit by myself and be in control of all my meanings.”

Her first published book was a collection of stories, Ophelia and the Great Idea, from 1988. “It’s a form I love to work in,” Levy says. “It’s very rigorous; you have to make a world very quickly, and you can play around with time.”

Asked if she shares the opinion of many short story writers that the form resembles poetry more than novels, Levy replies: “I think it’s more like film, because you have to set it up quickly: who wants what? Film is a very big influence on my writing, both avant-garde and mainstream, because there are so many strategies for visual storytelling. I love David Lynch, but then I love Spielberg; E.T. gets mentioned in Swimming Home. I always have to visualize before I write. I had to actually design the clinic in Hot Milk, get out pencil and paper so I could visualize it. There’s a metaphor there, because with the title I was thinking of maternal milk, and the clinic is shaped like a dome, a little bit like a breast. It’s made of a milky white marble that comes from marble quarries near the actual town.”

Set in the coastal town of Almería, Spain, Hot Milk replicates a strategy Levy employed to good effect in Swimming Home: assembling a group of people who don’t all know one another, in a location foreign to them. “I was really struck by the way writers like Henry James and E.M. Forster take their characters out of Britain and America and put them in Europe, in an unfamiliar place,” she explains. “It destabilizes them, and it really interests me to observe characters out of their comfort zone. With Hot Milk, I wanted to write about people in a burning desert landscape. Also, the game with my last two books has been that the darker the material and themes, the sunnier the skies will be. You know how literature students study the use of weather as an emotional metaphor? I do the opposite!”

Despite the rigor of her prose and the high-toned aura of her literary models, Levy in person is down-to-earth, with a wry wit and a ready laugh. Asked if she has a new novel underway, she declines to be specific, but says roguishly: “I reckon that we’re going to have to go for cold weather next time. I think I’m heading for a story set in the snow: thickly lined coats with velvet collars and hats and sturdy boots.”