  We introduce the concept of extrinsic catenary in the hyperbolic plane.
Working in the hyperboloid model, we define an extrinsic catenary as the shape
of a curve hanging under its weight as seen from the ambient space. In other
words, an extrinsic catenary is a critical point of the potential functional,
where we calculate the potential with the extrinsic distance to a fixed
reference plane in the ambient Lorentzian space. We then characterize extrinsic
catenaries in terms of their curvature and as a solution to a prescribed
curvature problem involving certain vector fields. In addition, we prove that
the generating curve of any minimal surface of revolution in the hyperbolic
space is an extrinsic catenary with respect to an appropriate reference plane.
Finally, we prove that one of the families of extrinsic catenaries admits an
intrinsic characterization if we replace the extrinsic distance with the
intrinsic length of horocycles orthogonal to a reference geodesic.
