Finiteness of local torsion for abelian t-modules

Let $C/\mathbb{F}_q$ be a regular projective curve, $\infty \in C$ a closed point, $A := \Gamma(C - \{\infty\}, \mathcal{O}_C)$, and $K := K(C)$ the fraction field of $A$. Consider a finite extension $L/K$, a place $v$ of $L$, and an abelian $A$-module $M$ (in the sense of Anderson) over $L_v$. We prove that the $L_v$-rational torsion submodule $M(L_v)_{\mathrm{tors}}$ of $M$ is a finite $A$-module.