Lipschitz spaces over non-porous sets

Let $M$ be a subset of ${\mathbb{R}}^n$. If $M$ is not porous, in particular if it has positive $n$-dimensional Lebesgue measure, we prove that the Lipschitz spaces ${\mathrm{L}\mathrm{i}\mathrm{p}}_0(M)$ and ${\mathrm{L}\mathrm{i}\mathrm{p}}_0({\mathbb{R}}^n)$ are linearly isomorphic. The result also holds more generally if ${\mathbb{R}}^n$ is replaced with a Carnot group equipped with its Carnot-Carathéodory metric.