  Given a constant $k>1$, let $Z$ be the family of round spheres of radius
$\textrm{artanh}(k^{-1})$ in the hyperbolic space $\mathbb{H}^3$, so that any
sphere in $Z$ has mean curvature $k$. We prove a crucial nondegeneracy result
involving the manifold $Z$. As an application, we provide sufficient conditions
on a prescribed function $\phi$ on $\mathbb{H}^3$, which ensure the existence
of a ${\cal C}^1$-curve, parametrized by $\varepsilon\approx 0$, of embedded
spheres in $\mathbb{H}^3$ having mean curvature $k +\varepsilon\phi$ at each
point.
