Lipschitz maps with prescribed local Lipschitz constants

Let Γ be a closed subset of a complete Riemannian manifold M of dimension ≥ 2, let f: M → N be a Lipschitz map to a complete Riemannian manifold N, and let ψ be a continuous function which dominates the local Lipschitz constant of f. We construct a Lipschitz map which agress with f on Γ and whose local Lipschitz constant is ψ.