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...
Best Answer
Two which are for food rather than cash:
Let $f = t^{2d} + f_1 t^{2d-1} + f_2 t^{2d-2}+ \cdots f_d t^d + \cdots+ f_2 t^2 +f_1 t + 1$ be a palindromic polynomial, so the roots of $f$ are of the form $\lambda_1$, $\lambda_2$, ..., $\lambda_d$, $\lambda_1^{-1}$, $\lambda_2^{-1}$, ..., $\lambda_d^{-1}$. Set $r_k = \prod_{j=1}^d (\lambda_j^k-1)(\lambda_j^{-k} -1)$.
Conjecture: The coefficients of $f$ are uniquely determined by the values of $r_1$, $r_2$, ... $r_{d+1}$.
Motivation: When computing the zeta function of a genus $d$ curve over $\mathbb{F}_q$, the numerator is essentially of the form $f$. (More precisely, it is of the form $q^d f(t/\sqrt{q})$ for $f$ of this form.) Certain algorithms proceed by computing the $r_k$ and recovering the coefficients of $f$ from them. Note that you have to recover $d$ numbers, so you need at least $r_1$ through $r_d$; it is known that you need at least one more and the conjecture is that exactly one more is enough.
Reward: Sturmfels and Zworski will buy you dinner at Chez Panisse if you solve it.
Consider the following probabilistic model: We choose an infinite string, call it $\mathcal{A}$, of $A$'s, $C$'s, $G$'s and $T$'s. Each letter of the string is chosen independently at random, with probabilities $p_A$, $p_C$, $p_G$ and $p_T$.
Next, we copy the string $\mathcal{A}$ to form a new string $\mathcal{D}_1$. In the copying process, for each pair $(X, Y)$ of symbols in $\{ A, C, G, T \}$, there is some probability $p_1(X \to Y)$ that we will miscopy an $X$ as a $Y$. (The $16$ probabilities stay constant for the entire copying procedure.)
We repeat the procedure to form two more strings $\mathcal{D}_2$ and $\mathcal{D}_3$, using new probability matrices $p_2(X \to Y)$ and $p_3(X \to Y)$.
We then forget the ancestral string $\mathcal{A}$ and measure the $64$ frequencies with which the various possible joint distributions of $\{ A, C, G, T \}$ occur in the descendant strings $(\mathcal{D}_1, \mathcal{D}_2, \mathcal{D}_3)$.
Our procedure depended on $4+3 \times 16$ inputs: the $(p_A, p_C, p_G, p_T)$ and the $p_i(X \to Y)$. When you remember that probabilities should add up to $1$, there are actually only $39$ independent parameters here, and we are getting $63$ measurements (one less than $64$ because probabilities add up to $1$). So the set of possible outputs is a semialgeraic set of codimension $24$.
Conjecture: Elizabeth Allman has a conjectured list of generators for the Zariski closure of the set of possible measurements.
Motivation: Obviously, this is a model of evolution, and one which (some) biologists actually use. Allman and Rhodes have shown that, if you know generators for the ideal for this particular case, then they can tell you generators for every possible evolutionary history. (More descendants, known sequence of branching, etc.) There are techniques in statistics where knowing this Zariski closure would be helpful progress.
Reward: Elizabeth Allman will personally catch, clean, smoke and ship an Alaskan Salmon to you if you find the generators. (Or serve it to you fresh, if you visit her in Alaska.)