Let $O_{K,S}$ be the ring of $S$-integers in a number field $K$. Dirichlet's unit theorem implies that the group of units in $O_{K,S}$ is a finitely generated group. In other words, the group $\mathbb G_{m}(O_{K,S})$ is finitely generated. In particular, if $T$ is a torus over $K$, then $T(O_{K,S})$ is finitely generated.

My question is whether this statement generalizes to other group schemes. [Edit: Daniel Litt's comment makes it clear that unipotent groups don't satisfy this finiteness property.]

**Question.** Let $G$ be a reductive group scheme over $O_{K,S}$. Is $G(O_{K,S})$ finitely generated?

The analogous question for proper group schemes has a positive answer by the Mordell-Weil theorem.

What if we restrict to semi-simple group schemes?