  Inspired by the work of Cherry, we introduce and study a new notion of Brody
hyperbolicity for rigid analytic varieties over a non-archimedean field $K$ of
characteristic zero. We use this notion of hyperbolicity to show the following
algebraic statement: if a projective variety admits a non-constant morphism
from an abelian variety, then so does any specialization of it. As an
application of this result, we show that the moduli space of abelian varieties
is $K$-analytically Brody hyperbolic in equal characteristic zero. These two
results are predicted by the Green-Griffiths-Lang conjecture on hyperbolic
varieties and its natural analogues for non-archimedean hyperbolicity. Finally,
we use Scholze's uniformization theorem to prove that the aforementioned moduli
space satisfies a non-archimedean analogue of the "Theorem of the Fixed Part"
in mixed characteristic.
