Beams of relativistic electrons and/or positrons leaving the surface of a strongly magnetized neutron star may give rise to gamma-ray bursts. The beams could be accelerated by strong, magnetically aligned electric fields that are produced by oscillations of the stellar surface. Here we investigate the particle acceleration in these electric fields, the resulting electron-positron pair cascade, and the gamma-ray emission. We find that beams of electrons and positrons moving parallel to the magnetic field are generated, with a reported differential energy distribution. These beams produce the bulk of the gamma-ray burst radiation below about 1 MeV by the resonant Compton scattering of thermal photons emitted from the stellar surface. The escaping synchrotron radiation from the cascade dominates the radiation spectrum above about 1 MeV.