This thought occurred to me earlier and I'm surprised I hadn't considered it previously. I get the feeling that no meaningful generalization can occur in a non-separable Banach space but on the surface it seems like a meaningful generalization can be had if the Banach space is separable. Even if it's possible, convergence and other issues likely severely complicate the matter.
Yes, it is possible. For example, Serge Lang exhibits the basics of a theory of differential forms on Banach manifolds in Chapter V of his Differential and Riemannian Manifolds. (Non-)Separability is not an issue.
According to Lang, a $p$-form on a Banach space $E$ is simply a continuous alternating $p$-linear map on $E$. This yields a notion of $p$-forms on any Banach manifold and Lang shows how to define exterior and Lie derivative and derives their fundamental algebraic properties in this context.