Let us work over $K = \mathbf{C}((t))$ for simplicity. We say that a smooth proper scheme $X/K$ has *good reduction* if it extends to a smooth and proper algebraic space $\mathcal{X}/\mathcal{O}_K$ where $\mathcal{O}_K = \mathbf{C}[[t]]$, and that it has *potentially good reduction* if for some finite extension $K' = \mathbf{C}((t'))$ ($t' = t^{1/N}$), the base change $X_{K'}$ has good reduction over $K'$.

My question may sound a bit silly:

Question.Can you give an example of a (smooth, projective) rational surface $X/K$ which does not have potentially good reduction?

On the one hand, the usual "homotopical" obstructions to good reduction vanish. The only nontrivial one I can think of is the action of the Galois group of $K$ on the Neron-Severi group ${\rm NS}(X_{\bar K})$, but that one factors through the action of a finite quotient, and hence becomes trivial over some finite extension $K'$.

On the other, it is easy to think of a possible culprit. Take a sufficiently complicated zero-dimensional subscheme $Z_0\subseteq \mathbf{P}^2_{\mathbf{C}}$, and pick a formal curve $$ z\colon \operatorname{Spec} \mathbf{C}[[t]] \longrightarrow \operatorname{Hilb}(\mathbf{P}^2_\mathbf{C}), \quad z(0) = [Z_0]$$ through $Z_0$ and such that $z(\eta)$ ($\eta$ being the generic point) corresponds to a smooth subscheme $Z_\eta \subseteq \mathbf{P}^2_K$. Take $X$ (resp. $\mathcal{X}$) be the blowup of $\mathbf{P}^2_K$ (resp. $\mathbf{P}^2_{\mathcal{O}_K}$) along $Z_\eta$ (resp. the subscheme $Z$ corresponding to $z$). Then $\mathcal{X}$ will not be smooth if $Z$ is complicated enough. Of course, this does not imply that $X$ does not have potentially good reduction, and I do not see how to check this.