The youngest of three children of Tobias William Tennant, son of the 2nd Baron Glenconner, and Lady Emma Cavendish, Tennant was a grandchild of Andrew Cavendish, 11th Duke of Devonshire and Deborah Mitford, one of the Mitford sisters. Tennant was born in London[citation needed] and raised on a sheep farm in the Scottish Borders. She attended St Leonards School in St Andrews and Marlborough College, then studied sculpture at Winchester School of Art.
In 1993, Tennant sent photographs of herself to a high fashion magazine, and a friend introduced her to fashion writer Plum Sykes (Victoria Rowland). She had recently had her nasal septum pierced, which was an unusual look for models at the time. Meisel then used her on the cover of Italian Vogue. Soon after, Karl Lagerfeld announced Tennant as the new face of Chanel, with an exclusive contract. He is reported to have thought her reminiscent of Coco Chanel. Priya Elan, writing in her Guardian obituary, credits her, together with Kate Moss and Erin O'Connor, with having "introduced an era of androgyny on the catwalk".
Tennant appeared on fashion catwalks for Shiatzy Chen, Bill Blass, Dior and Chanel. She was a muse of the designers Karl Lagerfeld, Gianni Versace, Alexander McQueen, Nicolas Ghesquière, Riccardo Tisci and Victoria Beckham. She also appeared in numerous other advertising campaigns, including Calvin Klein, Hermès, and in 2002 in an influential campaign for Burberry. In 2010, she starred in L.K.Bennett's Spring Summer campaign photographed by Tim Walker. During the Closing Ceremony of the 2012 London Olympics, Tennant along with Naomi Campbell and Kate Moss was one of the British models wearing fashions created by British designers specifically for the event.
In 2016, Tennant and Lady Isabella Cawdor premiered a new ready to wear collection they designed for the gunmaker and clothing retailer Holland & Holland. She continued to sculpt, and with her sister, a gilder, operated a luxury homewares company, Tennant & Tennant.
Tennant married French photographer and osteopath David Lasnet in her home village of Oxnam, Roxburghshire on 22 June 1999. They had four children together. The family lived near Duns, in the small village of Edrom in Berwickshire, also in the Scottish Borders. In August 2020, Tennant and Lasnet announced they had separated.
Tennant died suddenly on 22 December 2020, aged 50.