  This paper provides upper and lower bounds on the kissing number of congruent
radius $r > 0$ spheres in $\mathbb{H}^n$, for $n\geq 2$. For that purpose, the
kissing number is replaced by the kissing function $\kappa(n, r)$ which depends
on the radius $r$. After we obtain some theoretical lower and upper bounds for
$\kappa(n, r)$, we study their asymptotic behaviour and show, in particular,
that $\lim_{r\to \infty} \frac{\log \kappa(n,r)}{r} = n-1$. Finally, we compare
them with the numeric upper bounds obtained by solving a suitable semidefinite
program.
