  Consider a stationary Poisson process in a $d$-dimensional hyperbolic space.
For $R>0$ define the point process $\xi_R^{(k)}$ of exceedance heights over a
suitable threshold of the hyperbolic volumes of $k$th nearest neighbour balls
centred around the points of the Poisson process within a hyperbolic ball of
radius $R$ centred at a fixed point. The point process $\xi_R^{(k)}$ is
compared to an inhomogeneous Poisson process on the real line with intensity
function $e^{-u}$ and point process convergence in the Kantorovich-Rubinstein
distance is shown. From this, a quantitative limit theorem for the hyperbolic
maximum $k$th nearest neighbour ball with a limiting Gumbel distribution is
derived.
