applications of Berkovich spaces
I would first recommend the paper of Antoine Ducros (Espaces analytiques $p$-adiques au sens de Berkovich, Séminaire Bourbaki, exposé 958, 2006) for a general survey of the theory, with applications.
Here is a list of applications which I find striking, starting from those mentioned by Ducros's survey.
Étale cohomology. Berkovich developed a good theory of étale cohomology for his analytic spaces, which had applications in the Langlands program (for example, in the proof by Harris-Taylor of the local Langlands conjecture).
Proof (by Berkovich) of a conjecture of Deligne that the vanishing/nearby cycles (for a scheme over a discrete valuation ring) only depend on the formal completion.
Non-archimedean analogue of the classical potential theory on Riemann surfaces (Thuillier, Favre/Rivera-Letelier, Baker/Rumely).
Non-archimedean equidistribution theorems in the framework of Arakelov geometry (myself, Favre/Rivera-Letelier, Baker/Rumely, Gubler, Yuan), with applications to the Bogomolov conjecture for abelian varieties of function fields (Gubler, Yamaki), algebraic dynamics of Manin-Mumford/Mordell-Lang type (Yuan/Zhang, Dujardin/Favre,...).
Berkovich spaces of $\mathbf Z$ (Poineau) have applications to complicated rings of power series with integral coefficients introduced by Harbater and to their Galois theory. (In some sense, a geometrization of Harbater's formal patching.)
Mirror symmetry (Kontsevich/Soibelman) via the study of non-archimedean degenerations of Calabi-Yau manifolds. Recent developments in birational geometry (Mustață/Nicaise, Nicaise/Xu, Temkin) and viz. a non-archimedean analogue of the Monge-Ampère equation (Boucksom/Favre/Jonsson, Yuan/Zhang, Liu Y.).
Relation with tropical geometry (Baker/Payne/Rabinoff, my work with Ducros, Gubler/Rabinoff/Werner,...)
Relations with non-archimedean Arakelov geometry (Gubler/Künnemann, Ducros and myself)
A notable feature of the Berkovich spaces is the presence of (sometimes canonical) closed subspaces endowed with canonical piecewise linear structures on which the analytic spaces retracts by deformations (Berkovich, Hrushovski/Loeser,...). Those subspaces (“skeleta”) carry a large amount of geometric information and are of tremendous use in the theory.
Let me add a few more applications to what has already been mentioned.
Relation with Bruhat-Tits buildings (Berkovich, then Rémy/Thuillier/Werner). If $G$ is a reductive group over a non-archimedean valued field, then the Bruhat-Tits building $\mathscr{B}(G)$ of $G$ embeds into the analytification $G^{an}$ of $G$. If you choose a parabolic subgroup $P$, you have a map to $(G/P)^{an}$, which is the analytification of a proper variety, hence a compact space. This can help you compactify the building, describe the strata of the compactification, etc.
Complex dynamics. Favre and Jonsson used Berkovich spaces (actually some instance of it that they call "valuative tree") in order to study the dynamics of a polynomial endomorphism of $\mathbb{C}^2$ near a superattracting fixed point or at infinity.
Resolution of singularities (Thuillier). Let $X$ be an algebraic variety over a perfect field $k$ with an isolated singular point $x$. Let $f \colon Y \to X$ be a resolution of it such that $f^{-1}(x)$ is a normal crossing divisor $E$. Then the homotopy type of the incidence complex of $E$ is independent of the choice of the resolution. (Remark here that $k$ is any perfect field and that the Berkovich spaces that come up in the proof are over the field $k$ endowed with the trivial valuation.)
$p$-adic differential equations. André used Berkovich spaces in order to prove a conjecture of Dwork on the logarithmic growth of the solutions of $p$-adic differential equations (he actually needs Berkovich spaces only when the base field is not locally compact). In a different direction, Baldassarri (then Pulita and myself, and also Kedlaya) took up the study of $p$-adic differential equations on Berkovich curves. For instance, in my work with Pulita, we manage to give conditions for the finiteness of the de Rham cohomology of a curve with coefficients in some module with a connection. (Here, my point is that even if you can easily state the results in any theory you like, rigid geometry for instance, you will have a hard time proving them without Berkovich spaces.)
Jan Kiwi (Duke Journal) used Berkovich spaces to give the first proof of my conjecture that any sequence of quadratic rational maps which divergence in moduli space (of Mobius conjugacy classes) has at most two "rescaling limits".