A rigid analytic space $Y$ over a complete non-archimedean valued field $k$ is said to be of countable type if it has a countable (possibly finite) admissible covering by affinoids over $k$. 


Suppose that $X$ is an affinoid space over such a $k$ and $U$ is an admissible open subset of $X$ with respect to the strong $G$-topology on $X$ (see Non-Archimedean Analysis by Bosch, Guntzer, Remmert section 9.1.4). 


In section 4 of 'On one-dimensional separated rigid spaces'. Indag. Math. (N.S.) 6 (1995), no. 4, 439–451 by Q. Liu and M. van der Put there is given an example of such a pair $X$ and $U$ with $U$ not of countable type. However, I believe that example only works if the residue field is uncountable. My question is whether there is such an example if the residue field is itself countable. I suspect that there are examples but would be pleased if not. 



Are admissible open subsets of an affinoid space of countable type?
If the field admits a countable dense subfield is countable, the corresponding Berkovich space will be a metrizable topological space.
Therefore, every open subset is metrizable, hence paracompact,
hence each connected component will be countable at infinity.


