  We construct a derived enhancement of Hom spaces between rigid analytic
spaces. It encodes the hidden deformation-theoretic informations of the
underlying classical moduli space. The main tool in our construction is the
representability theorem in derived analytic geometry, which has been
established in our previous work. The representability theorem provides us
sufficient and necessary conditions for an analytic moduli functor to possess
the structure of a derived analytic stack. In order to verify the conditions of
the representability theorem, we prove several general results in the context
of derived non-archimedean analytic geometry: derived Tate acyclicity,
projection formula, and proper base change. These results also deserve
independent interest themselves. Our main motivation comes from non-archimedean
enumerative geometry. In our subsequent works, we will apply the derived
mapping stacks to obtain non-archimedean analytic Gromov-Witten invariants.
