This is a very broad question and it is difficult to know where to start. Remember that Berkovich's theory is a theory of analytic geometry, hence it makes sense to look for the counterpart of anything you have in complex analytic geometry: does there exist a good notion of Kähler manifold, for instance? I will try to be somehow more specific though and concentrate on the work of Berkovich as you suggest.
In the recent years, there were several attempts to understand the topology of Berkovich spaces better. In [Berkovich, Smooth $p$-adic analytic spaces are locally contractible, Inventiones, 1999], Berkovich proves that every smooth space is locally contractible. In [Hrushovski-Loeser, Non-archimedean tame topology and stably dominated types], they prove that the result holds for quasi-projective spaces. There are also some results by Thuillier, but, as far as I know, there are no written notes yet, and I am afraid that I cannot remember the exact level of generality he deals with. Anyway, I think that the question is open in full generality: are Berkovich spaces locally contractible?
In another direction, Berkovich has written a book called [Integration of One-forms on P-adic Analytic Spaces]. He basically constructs sheafs of primitives of one-forms and their iterates on smooth spaces. He proves that the resulting de Rham complex is exact in degree 0 and 1 but in higher degree the question is open.
Last, I would like to add a few words about Berkovich's paper [A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures]. The title says clearly what it is about. The weight zero subspace of some limit of mixed Hodge structures is given an interpretion using the Betti numbers of some Berkovich analytic space. It would certainly be very interesting to say similar things for higher weights.