Let $K$ be a local field of characteristic zero and $X$ an affinoid rigid space over $K$. Let $U\subset X$ be an affinoid subdomain, and consider a finite family of points $\{p_{1},\cdots, p_{n}\}\subset U$. Is it true that there is a rational subdomain $V\subset X$ such that $\{p_{1},\cdots, p_{n}\}\subset V\subset U$? The Theorem of Gerritzen-Grauert, Theorem 7.3.5 in the reference, implies that this is true for a single point. I think a result like this would make sense, but I don't see a straightforward way of proving it just from the statement of the theorem. Is there a known answer, or at least any literature related to this question?

*Bosch, S.; Güntzer, Ulrich; Remmert, Reinhold*, Non-Archimedean analysis. A systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften, 261. Berlin etc.: Springer Verlag. XII, 436 p. DM 168.00 (1984). ZBL0539.14017.