  Firstly, we pursue the work of W. Cherry on the analogue of the Kobayashi
semi distance dCK that he introduced for analytic spaces defined over a
non-Archimedean metrized field k. We prove various characterizations of smooth
projective varieties for which dCK is an actual distance. Secondly, we explore
several notions of hyperbolicity for a smooth algebraic curve X defined over k.
We prove a non-Archimedean analogue of the equivalence between having negative
Euler characteristic and the normality of certain families of analytic maps
taking values in X.
