  Let $R$ be a Banach ring. We prove that the category of chain complexes of
complete bornological $R$-modules (and several related categories) is a derived
algebraic context in the sense of Raksit. We then use the framework of derived
algebra to prove a version of the Hochschild-Kostant-Rosenberg Theorem, which
relates the circle action on the Hochschild algebra to the de
Rham-differential-enriched-de Rham algebra of a simplicial, commutative,
complete bornological algebra. This has a geometric interpretation in the
language of derived analytic geometry, namely, the derived loop stack of a
derived analytic stack is equivalent to the shifted tangent stack. Using this
geometric interpretation we extend our results to derived schemes.
