Let $k$ be a field equipped with non-Archimedean absolute value, let $S=\mathcal{M}(A)$ be an affinoid domain over $k$, and let $\pi_S: S^{an} \to \tilde{S}$ be the reduction map from the Berkovich analytification to reduction scheme $\mathrm{Spec}(A^\circ/A^{\circ\circ})$ with irreducible components $\tilde{S}_i$.

My question is to confirm the intuition coming from scheme theory that if some property holds over a generic point, then in holds over a generic open subset.

In particular, let $x \in S^{an}$ be such that $\pi_S(x)$ is a generic point of one of $\tilde{S}_i$. Denote by $\mathcal{H}(x)$ the completed residue field at the point $x$, which is a complete field with non-Archimedean absolute value. Let $X \to S, Y \to S$ be analytic morphisms and consider $X'= X \otimes_S \mathcal{H}(x), Y' \otimes_S \mathcal{H}(x)$. Assume that there exists an analytic map $X' \to Y'$. Does it then follow that there exists an affinoid subdomain $S' \subset S$, containing $x$, and an analytic map $X' \times_S S' \to Y' \times_S S'$?


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