If $N^2$ is a closed, orientable surface of genus at least $2$, and if $\phi$ is an (orientation-preserving) pseudo-Anosov mapping on $N$, then one can form the closed orientable 3-manifold $M^3$ by gluing the boundary components of $N^2\times [0,1]$ via $\phi$. In other words, $M^3$ is fibered over $S^1$, with fiber $N^2$. $M^3$ can also be given a hyperbolic structure (much like $N^2$ can). So my question is:

Does there exist a closed orientable hyperbolic manifold $M^k$, fibered over $S^1$ with a fiber $N^{k-1}$, when $k\neq 3$?

No such $M$ exists for $k=2$. It is also known (via Mostow rigidity) that for such an $M^k$ to exist for $k>3$, we cannot have the fiber $N^{k-1}$ hyperbolizable as well. But I don't know anything further than that.