Open problems in Berkovich geometry

I do not know if this falls within the scope of your question, and moreover I do not have a specific reference to point to, but there are certainly plenty of unsolved questions involving the dynamics of morphisms of Berkovich projective space $\mathbb{P}_{/\mathbb{C}_p}^{k,\mathrm{an}}$, particularly when $k > 1$, where not much is known. Here is one such open problem, which was related to me by Matt Baker.

Recall that to a morphism $f : \mathbb{C}\mathbb{P}^k \to \mathbb{C}\mathbb{P}^k$ of degree higher than one there is attached a canonical Green's $(1,1)$-current $T_f$, defined locally as the Laplacian of the escape rate function of any holomorphic lift $\mathbb{C}^{k+1} \setminus \{0\} \to \mathbb{C}^{k+1} \setminus \{0\}$, and then an equilibrium (Monge-Ampere) measure $\mu_f := T_f^{\wedge k}$, the self-intersection of $T_f$. When $k = 1$ and $f$ a polynomial this is the measure introduced by Lyubich in 1982; it is supported on the Julia set and reflects the distribution of periodic points, as well as of the backward orbit of an arbitrary non-exceptional point.

Now, the theory has a $p$-adic counterpart, and particular, any morphism of $f$ of $\mathbb{P}_{/\mathbb{C}_p}^{k,\mathrm{an}}$ (of degree higher than one) has attached to it a canonical measure $\mu_f$ on the space $\mathbb{P}_{/\mathbb{C}_p}^{k,\mathrm{an}}$. This measure was at first directly defined in [Chambert-Loir: "Mesures et equidistribution sur les espaces de Berkovich," Crelle, 2006], and the current $T_f$ was itself subsequently introduced and studied in [Chambert-Loir and Ducros: "Formes differentielles reelles et courants sur les espaces de Berkovich," 2012]. Prior to the latter paper, the definition of forms and currents on Berkovich spaces had itself been a major open problem.

The following open problem is I think rather interesting:

Conjecture. (Baker) The canonical measure $\mu_f$ charges a single point of $\mathbb{P}_{/\mathbb{C}_p}^{k,\mathrm{an}}$ if and only if $f$ has a good reduction after a linear change of coordinates.

When $k = 1$ a proof can be found in Baker and Rumely's book ("Potential Theory on the Berkovich Projective Line"). For higher dimension $k > 1$ this is open. The non-trivial direction is of course $\Rightarrow$ (the "only if").

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. Edit: I have just realized that, at the end of the introduction of the book, Berkovich gives a list of six open questions, the one I mentioned above being the first one. You may want to have a look at those.

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.

Edit: I have just realized that I forgot to say something about general Banach rings. Over an arbitrary Banach ring, almost nothing is done and it is even doubtful that one can actually do something in such a gereral setting.

There are some results over $\mathbb{Z}$ (or rings of integers of number fields): properties of local rings, mainly, but this is definitely only the very beginning of the theory. In particular, at the topological level (local arcwise connectedness, local contractibility, etc.) or at the cohomological level (finiteness of cohomology in the proper case, GAGA, etc.), there is nothing (yet). And here, I am only speaking of the usual transcendental topology and of the coherent cohomology. As far as I know, étale morphisms have not even been defined...