  Let $K$ be a complete algebraically closed non-archimedean valued field of
characteristic zero, and let $X$ be a finite type scheme over $K$. We say $X$
is $K$-analytically Borel hyperbolic if, for every finite type reduced scheme
$S$ over $K$, every rigid analytic morphism from the rigid analytification
$S^{\mathrm{an}}$ of $S$ to the rigid analytification $X^{\mathrm{an}}$ of $X$
is algebraic. Using the Viehweg-Zuo construction and the $K$-analytic big
Picard theorem of Cherry-Ru, we show that, for $N \geq 3$ and $g \geq 2$, the
fine moduli space $\mathcal{M}^{[N]}_{g,K}$ over $K$ of genus $g$ curves with
level $N$-structure is $K$-analytically Borel hyperbolic.
