  We construct nontrivial unbounded domains $\Omega$ in the hyperbolic space
$\mathbb{H}^N$, $N \in \{2,3,4\}$, bifurcating from the complement of a ball,
such that the overdetermined elliptic problem \begin{equation}
-\Delta_{\mathbb{H}^N} u+u-u^p=0\,\, \text{in}\,\,\Omega, \,\,
u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega\nonumber
\end{equation} has a positive bounded solution in
$C^{2,\alpha}\left(\Omega\right) \cap H^1\left(\Omega\right)$. We also give a
condition under which this construction holds for larger dimensions $N$. This
is linked to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined
elliptic problems, and, as far as we know, is the first nontrivial example of
solution to an overdetermined elliptic problem in the hyperbolic space.
