Isometries of ultrametric normed spaces

Abstract We show that the group of isometries an ultrametric normed space can be seen as a kind fractal. Then, we apply this description to study counterparts some classical problems in Archimedean analysis, such so called Problème des rotations de Mazur or Tingley’s problem. In particular, it turns out that, contrast with case real spaces, between spaces very far from being linear.