  The goal of the present text is to state and prove a generalization of
Raynaud localization theorem in the setting of derived geometry. More
explicitly, we show that the $\infty$-category of quasi-paracompact and
quasi-separated derived $k$-analytic spaces can be realized as a localization
of the $\infty$-category of admissible derived formal schemes. We construct a
derived rigidification functor generalizing Raynaud rigidification functor. In
order to construct the latter we will need to formalize derived formal
$\mathcal{O}_k$-adic formal geometry via a structured spaces approach. We prove
that $\mathcal{O}_k$-adic Postnikov towers of derived $\mathcal{O}_k$-adic
Deligne-Mumford stacks decompose and we relate these to Postnikov towers of
derived $k$-analytic spaces. This is possible by a precise comparison between
the $\mathcal{O}_k$-adic cotangent complex and the $k$-analytic cotangent
complex.

Algebraic Geometry