In the introduction to 'A convenient setting for Global Analysis', Michor & Kriegl make this claim: "The study of Banach manifolds per se is not very interesting, since they turnout to be open subsets of the modeling space for many modeling spaces."
But finite-dimensional manifolds are found to be interesting even though they can be embedded in some Euclidean space (of larger dimension). (Actually this seems to me, to make the above claim intuitively plausible, so that claim should be no more than we should expect).
But they do go on to say that "Banach manifolds are not suitable for many questions of Global Analysis, as … a Banach Lie group acting effectively on a finite dimensional smooth manifold it must be finite dimensional itself.", which does seem a rather strong limitation.
Best Answer
In his remarkable thesis Douady proved that, given a compact complex analytic space $X$, the set $H(X)$ of analytic subspaces of $X$ has itself a natural structure of analytic space .
If $X=\mathbb P^n(\mathbb C)$ for example, then $H(X)$ is the Hilbert scheme $ Hilb(\mathbb P^n(\mathbb C))$.
However the problem is much more difficult for non algebraic $X$.
Douady solved it by massive use of Banach analytic manifolds, the most important of them being the grassmannian of complemented closed subspaces of a Banach space.
The thesis starts with the candid statement of its aim: "Le but de ce travail est de munir son auteur du grade de docteur-ès-sciences mathématiques et l'ensemble H(X) des sous espaces analytiques compacts de X d'une structure d'espace analytique", that is to endow its author with the title of doctor in mathematics and the the set H(X) of compact analytic subspaces of X with the structure of analytic space.