  We present several results on the compactness of the space of morphisms
between analytic spaces in the sense of Berkovich. We show that under certain
conditions on the source, every sequence of analytic maps having an affinoid
target has a subsequence that converges pointwise to a continuous map. We also
study the class of continuous maps that arise in this way. Locally, they turn
analytic after a certain base change. We give some applications of these
results to the dynamics of an endomorphism f of the projective space. We define
the Fatou set as the normality locus of the family of the iterates {f n }. We
then generalize to the non-Archimedan setting a theorem of Ueda stating that
every Fatou component is hyperbolically imbedded in the projective space.
