Is there an "exponential law" for differentiable maps between smooth manifolds? Although it seems like a textbook question, I was not able to find a textbook or even a research article answering the following question: 
Let $M$, $N$ and $P$ be finite-dimensional smooth manifolds and let $f \in C^r(M \times N,P)$ for a given $r \in \mathbb{N}$. 
Let $\hat{f}: M \to C^0(N,P)$ denote the adjoint map, given by $\hat{f}(x) = (y \mapsto f(x,y))$.
For which $k,l \in \mathbb{N}$ does it hold that $\hat{f} \in C^k(M,C^l(N,P))$?
 A: As far as I am aware, Kriegl and Michor's book deal only with the smooth case. To my knowledge the first full account on an exponential law for finite orders of differentiability was given in 
Alzaareer, Schmeding: Differentiable mappings on products with different degrees of differentiability in the two factors, see https://arxiv.org/abs/1208.6510 for the preprint version.
there the linear case is dealt with. 
Let me spell this out here:
If $M,N$ are open subsets of finite dimensional spaces(actually one can do much better then open in fin. dim. space) and $P = \mathbb{R}^n$, then the adjoint construction written out by you yields a (linear) topological isomorphism 
$$C^k(M,C^l(N,P)) \rightarrow C^{k,l} (M\times N,P)$$
where the space on the right hand side consists of all mappings which are $k$-times continuously differentiable with respect to the $M$-variable and every one of these derivatives is $l$-times continuously differentiable with respect to the $N$-variable (the cited paper contains a careful study of these mappings, chain-rules etc.).
So for example, if in your notation $r\geq l+k$ you will always obtain a map in $C^k(M,C^l(N,P))$, as one sees that $C^{l+k} (M\times N, P) \subseteq C^{k,l} (M\times N, P)$.
Now this is only linear theory. However, one can easily generalise this to the manifold setting you want (by the chain rules in the cited paper it makes sense to talk about these mappings on manifolds). The theory then generalises to manifold valued mappings as follows: First of all we need that $N$ is a compact manifold (otherwise $C^l(N,P)$ will not be a Banach manifold with the compact open $C^l$-topology, or any other of the usual function space topologies for that matter). Then the straight forward calculations in local charts should yield the same answer as above in the linear case.
Addendum 1: The topology on the space $C^{k,l} (M\times N,P)$ is a variant of the compact open $C^r$-topology. Here one controls of course the derivatives up to order $k$ in the $M$ direction and up to order $l$ in the $N$ direction on compact subsets. This topology was also discussed in detail in the cited paper.
Addendum 2: Mappings of type $C^{k,l}$ are not new per se, for example in the literature on ODE's one can find these maps, e.g. in Amann: Ordinary differential equations, mappings of type $C^{0,1}$ are considered and in Treves: Topological Vector Spaces, Distributions and Kernels, one finds in a side remark something about the compatibility of $C^{r,s}$ mappings with topological tensor products. However, in neither source is a treatment of the topological properties of these spaces (whence the remark in Treves' book is more in the direction of "it should work for a suitably defined space").
