We establish integral analogues of results of Bushnell and Henniart for spaces of Whittaker functions arising from the groups GLn(F)GLn(F) for FF a pp-adic field. We apply the resulting theory to the existence of representations arising from the conjectural “local Langlands correspondence in families” and reduce the question of the existence of such representations to a natural conjecture relating the integral Bernstein center of GLn(F)GLn(F) to the deformation theory of Galois representations.