Non-archimedean analytification of algebraic spaces

We study quotient problems for étale equivalence relations in non-archimedean geometry, and we construct quotients for such equivalence relations in Berkovich's category of analytic spaces, assuming a separat-edness hypothesis on the equivalence relation. We also give counterex-amples that show the necessity of separatedness hypotheses, in contrast with the complex-analytic case. As an application, we construct ana-lytifications for separated algebraic spaces over a non-archimedean field.