The Ind-category of Banach spaces

Prosmans and Schneiders viewed the Ind-category of complex Banach spaces as a kind of idealized setting for sheaf theory on complex manifolds. It seems as if almost no one has taken up this approach since their work. I will review some features of the homological algebra of modules that can be done relative to this exact category. I will discuss how they can be helpful in formulating a theory of derived analytic geometry. To a large extent this machinery works when the complex numbers are replaced by a non-archimedean field or even the ring of integers with its usual absolute value. However, I will discuss one situation where things go wrong with this generalization and I do not know how to proceed.