  For a large class of metric spaces with nice local structure, which includes
Banach-Finsler manifolds and geodesic spaces of curvature bounded above, we
give sufficient conditions for a local homeomorphism to be a covering
projection. We first obtain a general condition in terms of a path continuation
property. As a consequence, we deduce several conditions in terms of
path-liftings involving a generalized derivative, and in particular we obtain
an extension of Hadamard global inversion theorem in this context. Next we
prove that, in the case of quasi-isometric mappings, some of there sufficient
conditions are also necessary. Finally, we give some applications to the
existence of global implicit functions.
