Lipschitz maps with prescribed local Lipschitz constants

Let $\mathrm{\Gamma }$ be a closed subset of a complete Riemannian manifold $M$ of dimension $\ge 2$, let $f:M\to N$ be a Lipschitz map to a complete Riemannian manifold $N$, and let $\psi$ be a continuous function which dominates the local Lipschitz constant of $f$. We construct a Lipschitz map which agress with $f$ on $\mathrm{\Gamma }$ and whose local Lipschitz constant is $\psi$.