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As Ryan points out, the interesting case is when the fiber is 2-dimensional. As Igor points out, this is a difficult open problem when the fiber has dimension 2.

When the fiber is a surface $F$, the fundamental group of the base $B$ admits a representation into the mapping class group $\mathrm{Mod}(F)$ of $F$. A combination of theorems of Farb-Mosher and Hamenstaedt shows that the fundamental group of the bundle is $\delta$-hyperbolic if and only if the map $\pi_1(B) \to \mathrm{Mod}(F)$ has finite kernel and the image is convex cocompact." (A subgroup of $\mathrm{Mod}(F)$ is convex cocompact if it acts on the Teichmueller space of $F$ with quasiconvex orbits.)

The only known examples of convex cocompact subgroups of mapping class groups are all virtually free, and so we are pretty far from knowing if there is a bona fide hyperbolic example like you are asking about. As Igor says, we don't even know if there is a $\delta$-hyperbolic surface-by-surface group.

However, it is conjectured that there is no hyperbolic (meaning truly hyperbolic) surface bundle $E$ over a surface. The reasoning is as follows: by an argument of Thurston, $E$ is symplectic. Using this fact, you can show that $E$ has a finite cover with nonvanishing Seiberg-Witten invariants. On the other hand, it is conjectured that the Seiberg-Witten invariants of a hyperbolic 4-manifold vanish. (See the article "Surface subgroups of mapping class groups" by Alan Reid in "Problems on Mapping Class Groups and Related Topics" edited by Benson Farb.)

For an introduction to convex cocompactness, and more references, see the survey "Subgroups of mapping class groups from the geometrical viewpoint" by me and Leininger.

Added: From the coarse perspective, the case where $F$ and $B$ are surfaces is key. If there were a $\delta$-hyperbolic $E$ with $F$ a surface and $B$ a hyperbolic manifold, then there is a $\delta$-hyperbolic $E'$ with fiber $F$ and base $B'$ a surface. To see this, note that recent work of Kahn-Markovic shows that $\pi_1(B)$ contains a quasiconvex surface group (I believe their theorem holds in all dimensions), and pulling back the bundle to this surface gives an example.

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As Ryan points out, the interesting case is when the fiber is 2-dimensional. As Igor points out, this is a difficult open problem when the fiber has dimension 2.

When the fiber is a surface $F$, the fundamental group of the base $B$ admits a representation into the mapping class group $\mathrm{Mod}(F)$ of $F$. A combination of theorems of Farb-Mosher and Hamenstaedt shows that the fundamental group of the bundle is $\delta$-hyperbolic if and only if the map $\pi_1(B) \to \mathrm{Mod}(F)$ has finite kernel and the image is convex cocompact." (A subgroup of $\mathrm{Mod}(F)$ is convex cocompact if it acts on the Teichmueller space of $F$ with quasiconvex orbits.)

The only known examples of convex cocompact subgroups of mapping class groups are all virtually free, and so we are pretty far from knowing if there is a bona fide hyperbolic example like you are asking about. As Igor says, we don't even know if there is a $\delta$-hyperbolic surface-by-surface group.

However, it is conjectured that there is no hyperbolic (meaning truly hyperbolic) surface bundle $E$ over a surface. The reasoning is as follows: by an argument of Thurston, $E$ is symplectic. Using this fact, you can show that $E$ has a finite cover with nonvanishing Seiberg-Witten invariants. On the other hand, it is conjectured that the Seiberg-Witten invariants of a hyperbolic 4-manifold vanish. (See the article "Surface subgroups of mapping class groups" by Alan Reid in "Problems on Mapping Class Groups and Related Topics" edited by Benson Farb.)

For an introduction to convex cocompactness, and more references, see the survey "Subgroups of mapping class groups from the geometrical viewpoint" by me and Leininger.