  We construct simply connected, complete, non-$CMC$ biconservative surfaces in
the $3$-dimensional hyperbolic space $\mathbb{H}^3$ in an intrinsic and
extrinsic way. We obtain three families of such surfaces, and, for each
surface, the set of points where the gradient of the mean curvature function
does not vanish is dense and has two connected components. In the intrinsic
approach, we first construct a simply connected, complete abstract surface and
then prove that it admits a unique biconservative immersion in $\mathbb{H}^3$.
Working extrinsically, we use the images of the explicit parametric equations
and a gluing process to obtain our surfaces. They are made up of circles (or
hyperbolas, or parabolas, respectively) which lie in $2$-affine parallel planes
and touch a certain curve in a totally geodesic hyperbolic surface
$\mathbb{H}^2$ in $\mathbb{H}^3$.
