  Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be
extended in a unique way to a bounded linear operator $\widehat{f} : \mathcal
F(M) \to \mathcal F(N)$ between their corresponding Lipschitz-free spaces. In
this paper, we give a necessary and sufficient condition for $\widehat{f}$ to
be compact in terms of metric conditions on $f$. This extends a result by A.
Jim\'{e}nez-Vargas and M. Villegas-Vallecillos in the case of non-separable and
unbounded metric spaces. After studying the behavior of weakly convergent
sequences made of finitely supported elements in Lipschitz-free spaces, we also
deduce that $\widehat{f}$ is compact if and only if it is weakly compact.
