I'm looking for an example of a rigid analytic space, over a field containing the $p$-adic numbers which is complete with respect to a non-archimedean norm, that is quasi-Stein, in the sense of Kiehl, but it not Stein or affinoid. Presumably such a space exists, I'm just not sure how to construct it or where to find such an example in the literature.
I would be particularly interested to hear about any spaces of the above form that appear in nature. This is deliberately vague: the sort of quasi-Stein, but not affinoid, spaces that I'm thinking of here are the open unit ball, rigid analytifications of affine varieties and the Drinfeld spaces. These are all Stein!
I have a suspicion that the rigid analytification functor might be a useful thing to consider here but I'm not sure.
Any help would be hugely appreciated!
Thanks, Tom
