Lehto--Virtanen-type and big Picard-type theorems for Berkovich analytic spaces

In non-archimedean setting, we establish a Lehto–Virtanen-type theorem for a morphism from the punctured Berkovich closed unit disk $\bar{\mathsf{D}}\setminus \{0\}$ in the Berkovich affine line to the Berkovich projective line ${\mathsf{P}}^1$ having an isolated essential singularity at the origin, and then establish a big Picard-type theorem for such an open subset $\mathrm{\Omega }$ in the Berkovich projective space ${\mathsf{P}}^N$ of any dimension $N$ that the family of all morphisms from $\bar{\mathsf{D}}\setminus \{0\}$ to $\mathrm{\Omega }$ is normal in a non-archimedean Montel’s sense. As an application of the latter theorem, we see a big Brody-type hyperbolicity of the Berkovich harmonic Fatou set of an endomorphism of ${\mathsf{P}}^N$ of degree $>1$.