The dynamics of energetic particles in strong electromagnetic fields can be heavily influenced by the energy loss arising from the emission of radiation during acceleration, known as radiation reaction. When interacting with a high-energy electron beam, today's lasers are sufficiently intense to explore the transition between the classical and quantum radiation reaction regimes. We present evidence of radiation reaction in the collision of an ultrarelativistic electron beam generated by laser-wakefield acceleration (epsilon > 500 MeV) with an intense laser pulse (a(0) > 10). We measure an energy loss in the postcollision electron spectrum that is correlated with the detected signal of hard photons (gamma rays), consistent with a quantum description of radiation reaction. The generated gamma rays have the highest energies yet reported from an all-optical inverse Compton scattering scheme, with critical energy epsilon(crit) > 30 MeV.