  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 Fr\'{e}chet spaces,
is included. The manifolds admit the Fisher-Rao metric and, unusually for the
non-parametric setting, Amari's $\alpha$-covariant derivatives for all
$\alpha\in R$. By construction, they are $C^\infty$-embedded submanifolds of
particular manifolds of finite measures. The statistical manifolds are dually
($\alpha=\pm 1$) flat, and admit mixture and exponential representations as
charts. Their curvatures with respect to the $\alpha$-covariant derivatives are
derived. The likelihood function associated with a finite sample is a
continuous function on each of the manifolds, and the $\alpha$-divergences are
of class $C^\infty$.
