Superconductivity without phonons has been proposed for strongly correlated electron materials that are tuned close to a zero-temperature magnetic instability of itinerant charge carriers1. Near this boundary, quantum fluctuations of magnetic degrees of freedom assume the role of phonons in conventional superconductors, creating an attractive interaction that 'glues' electrons into superconducting pairs. Here we show that superconductivity can arise from a very different spectrum of fluctuations associated with a local (or Kondo-breakdown) quantum critical point2, 3, 4, 5 that is revealed in isotropic scattering of charge carriers and a sublinear, temperature-dependent electrical resistivity. At this critical point, accessed by applying pressure to the strongly correlated, local-moment antiferromagnet CeRhIn5, magnetic and charge fluctuations coexist and produce electronic scattering that is maximal at the optimal pressure for superconductivity. This previously unanticipated source of pairing glue6 opens possibilities for understanding and discovering new unconventional forms of superconductivity.