Finite dimensional Lie groups have the nice property that if $V$ is a small neighborhood of the identity, and $U \subset V$ another neighborhood, then $V$ is covered by finitely many translates of $U$ (just because V is relatively compact).

Does the same property hold for small enough neighborhoods of the identity in an infinite dimensional Lie group? My favorite example is the group of compactly supported diffeomorphisms of $\mathbb{R}^n$ -- is it known whether it has this property?

Remark: I'm pretty sure this property is not equivalent to local compactness, so the fact that $Diff_c(\mathbb{R}^n)$ is not locally compact doesn't give an answer.