A space $X$ (by which I mean a CW complex) is acyclic if its reduced singular homology $\tilde H_\ast(X;\Bbb Z)$ is trivial in all degrees.
A discrete group $\pi$ is said to be acyclic if its classifying space $B\pi$ is acyclic.
A space $X$ is aspherical if its universal cover is contractible. A space $X$ is non-aspherical if its universal cover isn't contractible.
Question: Does there exist a non-aspherical, acyclic space $X$ whose fundamental group $\pi$ is also acyclic?