  We prove a large deviation principle for the point process of large Poisson
$k$-nearest neighbor balls in hyperbolic space. More precisely, we consider a
stationary Poisson point process of unit intensity in a growing sampling window
in hyperbolic space. We further take a growing sequence of thresholds such that
there is a diverging expected number of Poisson points whose $k$-nearest
neighbor ball has a volume exceeding this threshold. Then, the point process of
exceedances satisfies a large deviation principle whose rate function is
described in terms of a relative entropy. The proof relies on a fine
coarse-graining technique such that inside the resulting blocks the exceedances
are approximated by independent Poisson point processes.
