  In non-archimedean setting, we establish a Lehto--Virtanen-type theorem for a
morphism from the punctured Berkovich closed unit disk
$\overline{\mathsf{D}}\setminus\{0\}$ 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 $\Omega$ in the Berkovich projective space $\mathsf{P}^N$
of any dimension $N$ that the family of all morphisms from
$\overline{\mathsf{D}}\setminus\{0\}$ to $\Omega$ 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 $>1$.
