A characterisation of the Daugavet property in spaces of Lipschitz functions

We study the Daugavet property in the space of Lipschitz functions Lip0(M) on a complete metric space M. Namely we show that Lip0(M) has the Daugavet property if and only if M is a length metric space. This condition also characterises the Daugavet property in the Lipschitz free space F(M). Moreover, when M is compact, we show that either F(M) has the Daugavet property or its unit ball has a strongly exposed point. If M is an infinite compact subset of a strictly convex Banach space then the Daugavet property of Lip0(M) is equivalent to the convexity of M.