A striking signal of dark matter beyond the standard model is the existence of cores in the centre of galaxy clusters. Recent simulations predict that a brightest cluster galaxy ( BCG) inside a cored galaxy cluster will exhibit residual wobbling due to previous major mergers, long after the relaxation of the overall cluster. This phenomenon is absent with standard cold dark matter where a cuspy density profile keeps a BCG tightly bound at the centre. We test this hypothesis using cosmological simulations and deep observations of 10 galaxy clusters acting as strong gravitational lenses. Modelling the BCG wobble as a simple harmonic oscillator, we measure the wobble amplitude, A(w), in the BAHAMAS suite of cosmological hydrodynamical simulations, finding an upper limit for the cold dark matter paradigm of A(w) < 2 kpc at the 95 per cent confidence limit. We carry out the same test on the data finding a non-zero amplitude of A(w) = 11.82(-3.0)(+7.3) kpc, with the observations dis-favouring A(w) = 0 at the 3 sigma confidence level. This detection of BCG wobbling is evidence for a dark matter core at the heart of galaxy clusters. It also shows that strong lensing models of clusters cannot assume that the BCG is exactly coincident with the large-scale halo. While our small sample of galaxy clusters already indicates a non-zero A(w), with larger surveys, e.g. Euclid, we will be able to not only confirm the effect but also to use it to determine whether or not the wobbling finds its origin in new fundamental physics or astrophysical process.