  Let $\Omega$ be a domain in a smooth complete Finsler manifold, and let $G$
be the largest open subset of $\Omega$ such that for every $x$ in $G$ there is
a unique closest point from $\partial \Omega$ to $x$ (measured in the Finsler
metric). We prove that the distance function from $\partial \Omega$ is in
$C^{k,\alpha}_{loc}(G\cup \partial \Omega)$, $k\ge 2$ and $0<\alpha\le 1$, if
$\partial \Omega$ is in
  $C^{k,\alpha}$.
