The basic object of functional analysis is the topological vector space, so vector spaces with some topology, we can add additional structure by introducing metrics etc, but the underlying object is a linear space anyway.
I was wondering if there's any field of math that still studies functionals, but defined on manifolds instead of vector spaces.
I know, I am late to the game, but what you are really looking for are Banach manifolds.
These are geometric spaces that are locally modelled on Banach spaces much like smooth manifolds are modelled on $\mathbb{R}^n$ and hence are a non-linear version of Banach spaces. If the linear Banach space model is non-separable, such infinite-dimensional manifolds are even more interesting.
Furthermore, you can model infinite-dimensional non-linear spaces on all kinds of infinite-dimensional linear spaces, e.g. Frechet manifolds.