  Hyperbolic lattices are starting to be explored in search of novel phases of
matter. At the same time, non-Hermitian physics has come to the forefront in
photonic, optical, phononic, and condensed matter systems. In this work, we
introduce non-Hermitian hyperbolic matter and elucidate its exceptional
properties in depth. We use hyperbolic Bloch theory to investigate band
structures of hyperbolic lattices in the presence of non-Hermitian on-site gain
and loss as well as non-reciprocal hopping. Using various analytical and
numerical approaches we demonstrate widely accessible and tunable exceptional
points and contours in {10,5} tessellations, which we characterize using phase
rigidity, energy scaling, and vorticity. We further demonstrate the occurrence
of higher-order exceptional points and contours in the {8,4} tessellations
using the method of Newton polygons, supported by vorticity and phase rigidity
computations. Finally, we investigate the open boundary spectra and densities
of states to compare with results from band theory, along with a demonstration
of boundary localisation. Our results unveil an abundance of exceptional
degeneracies in hyperbolic non-Hermitian matter.
