Let $R$ be a (non-trivial) valuation ring of an algebraically closed field. Let $V$ be an integral affine scheme over $S=\mathrm{Spec} R$ and assume that $V(R)\neq \emptyset$ (so e.g. $V$ is faithfully flat over $S$).
It is well known (even under less assumptions) that $V$ is an inverse limit of finitely presented schemes over $S$. Assume that all the fibers of $V\to S$ are irreducible, can $V$ be approximated by finitely presented schemes over $S$ such that each have irreducible fibers?
