We report electron spin resonance studies of double perovskite Ba2CoWO6 single crystals. Above TN=14 K, we observe a paramagnetic resonance with geff≃3.66. Upon cooling, this mode transforms into a broad antiferromagnetic resonance (AFMR) with a zero-field energy gap of about 200 GHz. The AFMR becomes barely detectable at low magnetic fields, indicating unusually strong decay processes. We argue that symmetry-allowed anisotropic spin–spin interactions are a possible reason for the pronounced spin-wave damping. Linear spin-wave theory calculations support this scenario.