We discuss the new surprising observational results that indicate quite convincingly that the prompt emission of gamma-ray bursts (GRBs) is due to synchrotron radiation produced by a particle distribution that has a low-energy cut-off. The evidence of this is provided by the low-energy part of the spectrum of the prompt emission, which shows the characteristic F_ν ∝ ν^1/3 shape followed by F_ν ∝ ν^−1/2 up to the peak frequency. This implies that although the emitting particles are in fast cooling, they do not cool completely. This poses a severe challenge to the basic ideas about how and where the emission is produced, because the incomplete cooling requires a small value of the magnetic field to limit synchrotron cooling, and a large emitting region to limit the self-Compton cooling, even considering Klein–Nishina scattering effects. Some new and fundamental ingredient is required for understanding the GRBs prompt emission. We propose proton–synchrotron as a promising mechanism to solve the incomplete cooling puzzle.