Liquid-crystal elastomers (LCEs) are rubbers whose constituent molecules are orientationally ordered. Their salient feature is strong coupling between the orientational order and mechanical strain. For example, changing the orientational order gives rise to internal stresses, which lead to strains and change the shape of a sample. Orientational order can be affected by changes in externally applied stimuli such as light. We demonstrate here that by dissolving-rather than covalently bonding-azo dyes into an LCE sample, its mechanical deformation in response to non-uniform illumination by visible light becomes very large (more than 60 degrees bending) and is more than two orders of magnitude faster than previously reported. Rapid light-induced deformations allow LCEs to interact with their environment in new and unexpected ways. When light from above is shone on a dye-doped LCE sample floating on water, the LCE 'swims' away from the light, with an action resembling that of flatfish such as skates or rays. We analyse the propulsion mechanism in terms of momentum transfer.