Let $G$ be a group, and $M$ a connected compact smooth manifold. I'm studying $$ \pi_0 (map (BG,M)). $$

For $G$ a finite group, we know that this is trivial by the Sullivan conjecture on maps from classifying spaces which was proven by Miller. (This does not require smoothness of $M$.)

It is easy to construct a counterexample for infinite discrete groups (take $G=\mathbb{Z}$ and $M=S^1$).

Do we know anything if $G$ is a compact Lie group?

Let $G$ be a topological group, and $M$ a connected compact smooth manifold. I'm studying $$ \pi_0 (map (BG,M)). $$

For $G$ a finite group, we know that this is just a point by the Sullivan conjecture on maps from classifying spaces which was proven by Miller. (This does not require smoothness of $M$.)

On the other hand, if $G$ is an infinite discrete groups, this $\pi_0$ can be larger (take $G=\mathbb{Z}$ and $M=S^1$).

Question What happens if $G$ is a compact Lie group? Are there examples where this $\pi_0$ is more than a point?