We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class $C_b^k$ with respect to appropriate reference measures. The case $k=\infty$, in which the manifolds are modelled on Fréchet spaces, is included. The manifolds admit the Fisher-Rao metric and, unusually for the non-parametric setting, Amari's $α$-covariant derivatives for all $α\in R$. By construction, they are $C^\infty$-embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually ($α=\pm 1$) flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the $α$-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the $α$-divergences are of class $C^\infty$.