We report neutron scattering measurements of critical magnetic excitations in the weakly ferromagnetic superconductor UCoGe. The strong non-Landau damping of the excitations we observe, although unusual, has been found in another related ferromagnet, UGe2 at zero pressure. However, we also find that there is a significant anisotropy of the magnetic correlation length in UCoGe that contrasts with an almost isotropic length for UGe2. The values of the magnetic correlation length and damping are found to be compatible with superconductivity on small Fermi-surface pockets. The anisotropy may be important to explain why UCoGe is a superconductor at zero pressure while UGe2 is not.