Fréchet manifold structure on space of sections

I know that the space $mathsf{C}^infty(M;N)$ of smooth maps from a closed (smooth) manifold $M$ to a (smooth) manifold $N$ is a Fréchet manifold. I have been looking for a more general version of this statement along the following lines:

Let $p: E to B$ be a smooth fiber bundle, where $E$ and $B$ are manifolds with corners (with some additional assumptions on $p:E to B$?). Then $Gamma^infty(B;E) := { s: B to E mid s $ smooth, $ p circ s = mathsf{id}_B }$ is a Fréchet manifold (with corners?)

but I can’t seem to find a precise statement or proof of something like this anywhere. I’ve tried looking in "A Convenient Setting for Global Analysis," but that book seems to work in a very large amount of generality that is a bit beyond what I would need. The only generalizations I am looking for are:

• instead of functions $f: M to N$, we consider sections $s: B to E$ of a fiber bundle $p: E to B$,
• the manifolds in question can have boundary (or maybe even corners).

Then the original result for $mathsf{C}^infty(M;N)$ would then be recovered by taking $M$ and $N$ without corners and considering the trivial bundle $M times N to M$.

I would really appreciate it if anyone could suggest a reference where a result like this is stated/proven, or if someone could explain how I could formulate/prove this (namely, what are the charts on $Gamma^infty(B;E)$, what assumptions would we need on $p: E to B$, and do we need a notion of "Fréchet manifold with corners"?). Thanks very much!