Let $X$ be a connected proper smooth curve over a finite field (so the generic point of $X$ is the spectrum of a global field $K$), and let $G \rightarrow X$ be an affine $X$-group scheme of finite type. Is $G(X)$ finite?

I will accept any answer that assumes additional hypotheses on $G$, as long as the class of $G$ under consideration includes all reductive group schemes (i.e., smooth affine $G \rightarrow X$ whose geometric fibers are (connected) reductive groups). Note that separatedness of $G \rightarrow X$ ensures that after replacing $G$ by the schematic image of its generic fiber one may without loss of generality assume that $G$ is flat.