  Fermi gamma-ray telescope data of GRB 080916C with ~1e55 erg in apparent
isotropic gamma-ray energy, show a several second delay between the rise of 100
MeV - GeV radiation compared with keV - MeV radiation. Here we show that
synchrotron radiation from cosmic ray protons accelerated in GRBs, delayed by
the proton synchrotron cooling timescale in a jet of magnetically-dominated
shocked plasma moving at highly relativistic speeds with bulk Lorentz factor
Gamma ~ 500, could explain this result. A second generation electron
synchrotron component from attenuated proton synchrotron radiation makes
enhanced soft X-ray to MeV gamma-ray emission. Long GRBs with narrow, energetic
jets accelerating particles to ultra-high energies could explain the Auger
observations of UHE cosmic rays from sources within 100 Mpc for nano-Gauss
intergalactic magnetic fields. The total energy requirements in a proton
synchrotron model are proportional to Gamma^(16/3). This model for GRB 080916C
is only plausible if Gamma ~< 500 and the jet opening angle is ~ 1 degree.
