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.
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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.