A classical theorem is saying that every smooth 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 metrizable, hence paracompact. 
Is this true for all mapping spaces of the form $C^\infty (K,M)$ for $K$ compact?