Abstract: A Bruhat-Tits building can be embedded in the Berkovich analytic space associated to the underlying group. This was used by Berkovich to determine the automorphism group of Drinfeld's non-archimedean upper half-spaces. In this talk we consider analogs of Drinfeld's upper half-spaces over finite fields. We show that their automorphism group is the group of automorphisms of the ambient projective space. This is a problem in birational geometry, which we solve with tools from non-archimedean analytic geometry and by considering a suitable spherical building.