  We prove that if a metric space $M$ has the finite CEP then $\mathcal
F(M)\widehat{\otimes}_{\pi} X$ has the Daugavet property for every non-zero
Banach space $X$. This applies, for instance, if $M$ is a Banach space whose
dual is isometrically an $L_1(\mu)$ space. If $M$ has the CEP then $L(\mathcal
F(M),X)=\Lip(M,X)$ has the Daugavet property for every non-zero Banach space
$X$, showing that this is the case when $M$ is an injective Banach space or a
convex subset of a Hilbert space.
