There is a paper of Lott and Villani that defines the concept of nonnegative Ricci curvature in an arbitrary measured length space:
There is a dimension parameter which can be taken to infinity in that definition.
In his proof of the Poincare and geometrisation conjectures, Perelman used a heuristic argument in which he formally applied the Bishop-Gromov inequality to an infinite-dimensional, formally Ricci flat manifold to obtain the monotonicity of what is now known as the Perelman reduced volume; see Section 6 of
or my notes on this at
EDIT: Some formal computations of Ricci and Riemann curvature on the (infinite dimensional) space of volume-preserving diffeomorphisms also appear in a famous paper of Arnold: