Why are Banach manifolds not so popular?
Solution 1:
They are popular (in the sense of: people would like to use them, some methods of dealing with nonlinear PDE are based on the theory, for example), but in some respects more difficult to handle than the finite dimensional manifolds. B-Space can be, e.g., non-reflexive, which requires additional care when you want to study the cotangent space. Also, in infinite dimensions, compactness (which is one highly important ingredient in existence proofs) is a rather complex issue. This means, in particular, that they are often only treated/used in advanced courses.