It seems that when we talk about the $p$-adic uniformization, we typically mean those uniformized by either the Drinfel'd upper spaces (for which we think of the examples of Mumford curves and some local studies of Shimura varieties), or the Tate uniformizaation of abelian varieties with multiplicative reduction (which we roughly think of as uniformized by tori). I would like to know what other kinds of spaces have been used in $p$-adic uniformization other than these two cases, as I haven't yet found much in the literature.
And in general, what kind of spaces could one expect to appear in the $p$-adic uniformization as the "universal covering" spaces? what is the $p$-adic counterpart of "simply connected" spaces? Since smooth Berkovich analytic spaces are locally contractible, can we expect such spaces exist in the sense of Berkovich?