  Let us consider a Riemannian manifold $M$ (either separable or
non-separable). We prove that, for every $\epsilon>0$, every Lipschitz function
$f:M\rightarrow\mathbb R$ can be uniformly approximated by a Lipschitz,
$C^1$-smooth function $g$ with $\Lip(g)\le \Lip(f)+\epsilon$. As a consequence,
every Riemannian manifold is uniformly bumpable. The results are presented in
the context of $C^\ell$ Finsler manifolds modeled on Banach spaces. Sufficient
conditions are given on the Finsler manifold $M$ (and the Banach space $X$
where $M$ is modeled), so that every Lipschitz function $f:M\rightarrow \mathbb
R$ can be uniformly approximated by a Lipschitz, $C^k$-smooth function $g$ with
$\Lip(g)\le C \Lip(f)$ (for some $C$ depending only on $X$). Some applications
of these results are also given as well as a characterization, on the separable
case, of the class of $C^\ell$ Finsler manifolds satisfying the above property
of approximation. Finally, we give sufficient conditions on the $C^1$ Finsler
manifold $M$ and $X$, to ensure the existence of Lipschitz and $C^1$-smooth
extensions of every real-valued function $f$ defined on a submanifold $N$ of
$M$ provided $f$ is $C^1$-smooth on $N$ and Lipschitz with the metric induced
by $M$.
