  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.
