Suppose that $X$ is a compact, finite dimensional manifold and $Y$ is an infinite dimensional, second countable ($C^\infty$-)Banach manifold. Let $\nu \in \mathbb{N}$.

Question:Is the space $C^\nu(X,Y)$ a ($C^\infty$ or $C^k$ for some $k \in \mathbb{N}$) Banach manifold which isLindelöf(i.e. every open cover has a countable subcover)?

Any help, as well as references are very much appreciated.

Some motivation:

My goal is to apply a parametric transversality theorem, where my parameter space is $C^\nu(X,Y)$. In order to so it is required that the parameter space is Lindelöf and a ($C^k$-) Banach manifold. I don't have much experience with function spaces and their topology, especially when the target is infinite dimensional. So I apologize if it turns out that my question is "well known".