Pełczyński's property (V${}^{\ast }$) in Lipschitz-free spaces

We prove that Pelczyński’s property (V${}^{\ast }$) is locally determined for Lipschitz-free spaces, and obtain several sufficient conditions for it to hold. We deduce that $\mathcal{F}(M)$ has property (V${}^{\ast }$) when the complete metric space $M$ is locally compact and purely 1-unrectifiable, a Hilbert space, or belongs to a class of Carnot-Carathéodory spaces satisfying a bi-Hölder condition, including Carnot groups.