  Derived mapping stacks are a fundamental source of examples of derived
enhancements of classical moduli problems. For instance, they appear naturally
in Gromov-Witten theory and in some branches of geometric representation
theory. In this paper, we show that in many cases the mapping stacks
construction commutes with the (complex or non-archimedean) analytification
functor. Along the way, we establish several properties of the stack of
analytic perfect complexes and study some incarnations of analytic Tannaka
duality.
