Let $K$ be a $p$-adic field and let $X$ be the rigid space $ \operatorname{Max} K\langle T_1, T_2 \rangle$, i.e. the 2-dimensional closed unit ball.

Consider the sets $U := \{ |T_1| < 1\}$ and $V := \{ |T_2| = 1\}$. It's standard that $U$ and $V$ are both admissible open subsets of $X$ (for the strong $G$-topology); e.g. this follows from Propositions 9.1.4/4 and 9.1.4/5 of Bosch--Guentzer--Remmert. However, it's not part of the axioms for a G-topology that the union of two admissible open sets be admissible open.

- Is $U \cup V$ an admissible open set for the $G$-topology of $X$?
~~If so, is $\{ U, V\}$ an admissible covering of $U \cup V$?~~

EDIT: I just realised these things can't both be true. Since admissible coverings are by definition stable under pullback by morphisms of affinoids, if $U \cup V$ is admissible and admissibly covered by $U$ and $V$, then the same has to be true after intersecting with the diagonal $Z = \{ T_1 = T_2\} \cong \operatorname{Max} K\langle T \rangle$. But intersecting with $Z$ gives precisely the canonical example of a non-admissible covering, $\{ |T| = 1\} \cup \{ |T| < 1\}$.

So either $U \cup V$ is not an admissible open, or it is admissible but the covering $\{U, V\}$ of it is not an admissible covering. I'd still like to know which of these is true.