You can't find a reference because it's false.

Take your favorite 'bad', but compact set $C \subset \mathbf{R}$, and define $f: x \in \mathbf{R} \mapsto \operatorname{dist}(x,C)$. This is $1$-Lipschitz, but not differentiable at the boundary points of $C$, which may well be large. For instance, you may take $C$ to be the Cantor set.