Recent observations of galaxy cluster Abell 3827 reveal a spatial offset between the galaxies and their associated dark matter haloes. It has been suggested that dark matter self-interactions of strength 1.0cm^2 /g<~σ/m<~1.5cm^2 /g could be responsible for the observed discrepancies. We propose a simple model for the dark sector consisting of a Dirac fermion and a scalar mediator. This dark sector is allowed to communicate with the Standard Model through mixing of the scalar mediator with the Higgs. Examination of the dark matter’s thermal freeze-out from p-wave annihilation, in combination with a theoretical upper bound placed on the mediator-fermion coupling strength, indicates that in order to satisfy the observations of Abell 3827, the mediator must be extremely light—between 0.69MeV and 100MeV—whereas the dark matter fermion could have a mass from 0.35GeV to 8200GeV. Analysis of the Higgs-mediator mixing reveals an allowed range of mediator lifetimes, from 3.2×10^(-10)s to 10^(-2)s.