Superradiance is a collective, coherent emission of photons from an excited ensemble of emitters, without any mediation by a photonic resonator or dipole-rephasing mechanism. In the regime where an atomic ensemble's superradiant behaviour enables photonic emission on timescales faster than the atoms' natural lifetime, we experimentally demonstrate a fast quantum memory protocol for broadband photonic signals in a cloud of laser-cooled rubidium atoms. We analyze the conditions for optimal operation of a superradiance-mediated memory, and show that for a given optical depth, this protocol offers the highest-bandwidth storage among protocols in the same system. Highbandwidth superradiance quantum memories provide unique opportunities for fast processing of optical and microwave photonic signals, with applications in large-scale quantum communication and quantum computing technologies.