Smooth classifying spaces for differential $K$-theory

We construct a version of differential $K$-theory based on smooth Banach manifold models for the homotopy types $B \mathrm U\times Z$ and $\mathrm U$ that appear in the topological $K$-theory spectrum. These manifolds carry natural differential forms that refine the topological universal Chern character, together with natural addition and inversion operations that induce the respective structure on $\hat K$. Our models are norm completions of the usual stable Grassmannian and the stable unitary group. Their regularity allows us to work completely on the level of classifying spaces, and therefore we do not need a compactness assumption on our manifolds that is present in many other descriptions. The constructed groups $\hat K(M)$ are isomorphic to the unique differential extension of $K$-theory that admits an $S^1$-integration.