  We introduce and study a new random surface which we call the hyperbolic
Brownian plane and which is the near-critical scaling limit of the hyperbolic
triangulations constructed in arXiv:1401.3297. The law of the hyperbolic
Brownian plane is obtained after biasing the law of the Brownian plane
arXiv:1204.5921 by an explicit martingale depending on its perimeter and volume
processes studied in arXiv:1409.4026. Although the hyperbolic Brownian plane
has the same local properties as those of the Brownian plane, its large scale
structure is much different since we prove e.g. that is has exponential volume
growth.
