Isometries, Mazur–Ulam theorem and Aleksandrov problem for non-Archimedean normed spaces

We study isometries between normed spaces over a non-Archimedean valued field KK. We show the failure of a Mazur–Ulam theorem in the framework of non-Archimedean spaces. Considering Aleksandrov problem, we prove that a surjective Lipschitz map E→EE→E with the strong distance one preserving property, where EE is a finite-dimensional normed space, is an isometry if and only if KK is locally compact. We prove also that every isometry E→EE→E for finite-dimensional EE is surjective if and only if KK is spherically complete and card(k) is finite.