Desert Island Duke by Kate Bateman

Surviving on a tropical island will be hard.
Living with him will be impossible. . .
Shipwrecked on a tropical isle, Caroline ‘Caro’ Montgomery is furious to discover that her sole fellow castaway is her brother’s infuriating best friend, Maximillian Cavendish, the fourteenth duke of Hayworth.
Max might be the most handsome man she’s ever met, but he’s teased her ever since they were children. So when Max claims to have lost his memory, Caro seizes the opportunity for revenge—and informs him he’s a lowly stable hand.
As they work together to survive, attraction flares hot between them. But should Caro reveal her deception? And what will happen if Max finally remembers he’s a duke?