A classical theorem is saying that every smooth, finite-dimensional manifold has a smooth partition of unity. My question is:

1. Which Fréchet manifolds have a smooth partition of unity?

2. How is the existence of smooth partitions of unity on Fréchet manifolds related to paracompactness of the underlying topology?

From some remarks in some literature, I got the impression that not all Fréchet manifolds have smooth partitions of unity, but some have, e.g. the loop space $LM$ of a finite-dimensional smooth manifold $M$.

For $LM$, the proof seems to be that $LM$ is Lindelöf, hence paracompact. Is this true for all mapping spaces of the form $C^\infty (K,M)$ for $K$ compact?