  We propose a derived version of non-archimedean analytic geometry.
Intuitively, a derived non-archimedean analytic space consists of an ordinary
non-archimedean analytic space equipped with a sheaf of derived rings. Such a
naive definition turns out to be insufficient. In this paper, we resort to the
theory of pregeometries and structured topoi introduced by Jacob Lurie. We
prove the following three fundamental properties of derived non-archimedean
analytic spaces:
  (1) The category of ordinary non-archimedean analytic spaces embeds fully
faithfully into the $\infty$-category of derived non-archimedean analytic
spaces.
  (2) The $\infty$-category of derived non-archimedean analytic spaces admits
fiber products.
  (3) The $\infty$-category of higher non-archimedean analytic Deligne-Mumford
stacks embeds fully faithfully into the $\infty$-category of derived
non-archimedean analytic spaces. The essential image of this embedding is
spanned by $n$-localic discrete derived non-archimedean analytic spaces.
  We will further develop the theory of derived non-archimedean analytic
geometry in our subsequent works. Our motivations mainly come from intersection
theory, enumerative geometry and mirror symmetry.
