Galaxy cluster Abell 3827 hosts the stellar remnants of four almost equally bright elliptical galaxies within a core of radius 10 kpc. Such corrugation of the stellar distribution is very rare, and suggests recent formation by several simultaneous mergers. We map the distribution of associated dark matter, using new Hubble Space Telescope imaging and Very Large Telescope/Multi-Unit Spectroscopic Explorer integral field spectroscopy of a gravitationally lensed system threaded through the cluster core. We find that each of the central galaxies retains a dark matter halo, but that (at least) one of these is spatially offset from its stars. The best-constrained offset is 1.62(-0.49)(+0.47) kpc, where the 68 per cent confidence limit includes both statistical error and systematic biases in mass modelling. Such offsets are not seen in field galaxies, but are predicted during the long infall to a cluster, if dark matter self-interactions generate an extra drag force. With such a small physical separation, it is difficult to definitively rule out astrophysical effects operating exclusively in dense cluster core environments - but if interpreted solely as evidence for self-interacting dark matter, this offset implies a cross-section sigma(DM)/(m) similar to (1.7 +/- 0.7) x 10(-4) cm(2) g(-1) x (t(infall)/10(9) yr)(-2), where t(infall) is the infall duration.