  We construct, for any $H\in \mathbb{R}$, infinitely many free boundary annuli
in geodesic balls of $\mathbb{S}^3$ with constant mean curvature $H$ and a
discrete, non-rotational, symmetry group. Some of these free boundary CMC
annuli are actually embedded if $H\geq 1/\sqrt{3}$. We also construct embedded,
non-rotational, free boundary CMC annuli in geodesic balls of $\mathbb{H}^3$,
for all values $H>1$ of the mean curvature $H$.
