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

Derived $\mathcal{O}_k$-adic geometry and derived Raynaud localization theorem