I have read that $H^i(K(\mathbb{R},1)$) has rank $2^\omega$ for any $i\in \mathbb{N}$ (see Thurston's comment here Nontrivial finite group with trivial group homologies?) therefore $K(\mathbb{R},1)$ is not a finite CW complex and even the $n$-skeleton is not finite for any $n$.

On the other hand there are infinite (discrete) Lie groups $\pi$, such that $K(\pi,1)$ is finite. For example consider $\pi = \pi_1 (E)$ where $E\subset \mathbb{S}^3$ is the knot exterior of a knot.
This seems to be a rather lucky case as we know that $K(F,1)$ is non finite for any discrete *finite* group $F$.

This made me wonder if the following is true:

Is $K(G,1)$ an infinite CW complex for any $G$ Lie group of dimension greater than 1?

What are other examples of $G$ such that $K(G,1)$ is finite?

Note: infinite CW complex is the same as being non compact.