Examples of Banach manifolds with function spaces as tangent spaces I have recently been learning the theory of Banach manifolds through Serge Lang's book on Differential Manifolds. So far the objects seem rather interesting but my intuition always comes from the finite dimensional case, which admittedly is very helpful but I certainly want to add onto this.
I am wondering if there is a reference where one has examples of Banach manifolds where the tangent spaces are function spaces (ie Sobolev spaces). Obviously, I would like to avoid the trivial example of the function space being the manifold.
In some sense I would like an example that embraces the infinite dimensional tangent space as well as giving me some understanding as to what the topological space/smooth structure should be in the setting.
 A: This doesn't exactly fit your criteria as it's not a Banach manifold,  but one good example of an infinite dimensional space is the space of probability measures with finite second moment along with the formal Riemannian metric induced by Otto Calculus
There are many ways to think of the tangent space. The natural coordinates are to consider all signed measures with finite second moment and zero total mass. However, one of Otto's key insights was that if you "change coordinates" by solving a particular elliptic differential equation (whose coefficients depend on the base point), there is a natural inner product that one can define
You have to be careful with the functional analysis to make this rigorous, and I believe the space  is actually fairly singular (which is why it's merely formal). However, this is a really important construction since the inner product induces the 2-Wasserstein metric as its distance function, so provides a bridge between optimal transport and functional analysis. Just to give  one example of an insight this provides, if you use this metric the heat equation is the gradient flow of the entropy.
A: Banach manifolds as a natural generalization of the finite dimensional case became “popular” in the late 60s. In Physics an obvious example would be the projective Hilbert space, the state space. Klingenberg’s book on Riemannian Geometry has some chapters on (Hilbert) Riemannian Geometry. Infinite Hamiltonian systems and symplectic geometryare treated extensively by many authors (E.G. Marsden,  Ratiu, Weinstein etc.).
However results by Eells/Elworthy 1970, Elworthy 1972, Henderson 1969, 1972, show that Hilbert manifolds are essentially open subsets of their tangent space and smooth Banach manifolds are more or less all diffeomorphic (Henderson 1972).
Also it would appear natural to look at the diffeomorphism group of a finite dimensional manifold. The tangent space should be given by the space of vector fields. Unfortunately a result by (Omori 1970) shows Banach manifolds are not suitable in that important case, more general (topological) vector spaces are needed. See the introduction of Kriegl/Michor 1997: A Convenient Setting of Global Analysis.
