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 \(H\ge 1/\sqrt{3}\) . 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.
