  It is now a classical result that an algebraic space locally of finite type
over $\mathbf{C}$ is analytifiable if and only if it is locally separated. In
this paper we study non-archimedean analytifications of algebraic spaces. We
construct a quotient for any etale non-archimedean analytic equivalence
relation whose diagonal is a closed immersion, and deduce that any separated
algebraic space locally of finite type over any non-archimedean field $k$ is
analytifiable in both the category of rigid spaces and the category of analytic
spaces over $k$. Also, though local separatedness remains a necessary condition
for analytifiability in either of these categories, we present many surprising
examples of non-analytifiable locally separated smooth algebraic spaces over
$k$ that can even be defined over the prime field.
