Let $M$ and $N$ be (finite-dimensional) smooth manifolds without boundary. For simplicity, assume $M$ is compact. This post concerns spaces of "copies of $M$ in $N$". To the best of my knowledge, the following are true:
Fact 1: Let $\mathcal{I}^\infty$ denote the space of $C^\infty$ embeddings $M\hookrightarrow N$. Then $\mathcal{I}^\infty$ is a smooth Fréchet manifold. For an embedding $i\in\mathcal{I}^\infty$, the tangent space $T_i\mathcal{I}^\infty$ can be thought of as the space of $C^\infty$ vector fields along $i$, which is indeed a Fréchet space.
Fact 2: Let $\mathcal{S}^\infty$ denote the space of smooth submanifolds of $N$ diffeomorphic to $M$. Then $\mathcal{S}^\infty$ is also a smooth Fréchet manifold. For a submanifold $S\in\mathcal{S}^\infty$, the tangent space $T_S\mathcal{S}^\infty$ can be thought of as the space of $C^\infty$ sections of the normal bundle.
Fact 3: Let $k$ be a positive integer, and let $\mathcal{I}^k$ denote the space of $C^k$ embeddings $M\hookrightarrow N$. Then $\mathcal{I}^k$ is a smooth Banach manifold. A tangent space $T_i\mathcal{I}^k$ can be thought of as the space of $C^k$ vector fields along $i$. This space is Banach.
And finally,
My question: Let $\mathcal{S}^k$ denote the space of $C^k$ submanifolds of $N$ which are ($C^k$) diffeomorphic to $M$. Is $\mathcal{S}^k$ a smooth Banach manifold? If not, is it a $C^l$ Banach manifold for some $l$?
I can't think of any reason why $\mathcal{S}^k$ shouldn't be a Banach manifold. At least on the tangent-space level, everything seems to work well. Indeed, let $S\in\mathcal{S}^k$. Then the space of $C^k$ vector fields on $S$ is a closed subspace of the space of $C^k$ vector fields along the inclusion $S\hookrightarrow N$, which is Banach. So, the space of $C^k$ sections of the normal bundle is also Banach (as the quotient of a Banach space by a closed subspace) and seems to be a reasonable model for the tangent space $T_S\mathcal{S}^k$. However, this is certainly not a sufficient answer to the question. Furthermore, I know that in some cases, arguments which work perfectly for smooth maps fail to hold for $C^k$ maps. Therefore, I wouldn't be surprised to hear that $\mathcal{S}^k$ does not have a natural smooth structure.
I will be grateful for any insight, as well as a reference to a text containing a clear discussion of this matter.