  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.
