Let $A$ be a DVR with an algebraically closed residue field $k$. Consider a morphism $f: X \to Y$ of arithmetic surfaces (regular integer projective and flat schemes over $A$ of dimension 2), such that $f$ induces a finite morphism on generic fibers. Assume that $X$ and $Y$ are semistable, the generic fibers of $X$ and $Y$ are smooth, and $Y$ is relatively minimal, i.e. does not contain any exceptional divisor.
Q: Is there any chance that $f$ factors through a birational map $X \to X'$, where $X'$ is a relatively minimal surface?
As far I understand, the answer lies in the consideration of an irreducible component $C$ of the closed fiber $X_k$ of $X$, which is isomorphic to $\mathbb P^1_k$ and intersects the other components of $X_k$ in exactly one point. With respect to the morphism $X \to X'$ the component $C$ is then contracted. If $f$ contracts $C$, so $f$ can possibly factor through $X'$. But in general (it seems to me) it could happen, that $f(C)$ is irreducible and intersects the other components of $Y_k$ in at least two points.