  We propose an extension of the Chronological Calculus, developed by Agrachev
and Gamkrelidze for the case of $C^\infty$-smooth dynamical systems on
finite-dimensional $C^\infty$-smooth manifolds, to the case of $C^m$-smooth
dynamical systems and infinite-dimensional $C^m$-manifolds. Due to a relaxation
in the underlying structure of the calculus, this extension provides a powerful
computational tool without recourse to the theory of calculus in Fr\'echet
spaces required by the classical Chronological Calculus. In addition, this
extension accounts for flows of vector fields which are merely measurable in
time. To demonstrate the utility of this extension, we prove a variant of
Chow-Rashevskii theorem for infinite-dimensional manifolds.
