Intriguing sub-TeV tails in the pulsed γ-ray emission from the Crab pulsar have recently been discovered by the Major Atmospheric Gamma-Ray Imaging Cherenkov (MAGIC) telescope and Very Energetic Radiation Imaging Telescope Array System (VERITAS) collaborations. These were not clearly predicted by any pulsar model. At present, it is argued that this emission is produced by electrons in the inverse Compton process that occur either in the outer gap of the pulsar magnetosphere or in the pulsar wind region at some distance from the light cylinder. We analyse another scenario, which is consistent with the basic features of this enigmatic emission. It is proposed that this emission is caused by electrons accelerated very close to the light cylinder where the e± plasma cannot saturate the induced huge electric fields. Electrons reach energies sufficient for the production of hard γ-ray spectra in the curvature radiation process. Because of different curvature radii of the leading and trailing magnetic field lines, the γ-ray spectra from separate pulses should extend to different maximum energies. The scenario can also explain the lower-level γ-ray emission from the interpulse region (between P1 and P2) observed in the light curve of the Crab pulsar. Moreover, we argue that pulsars with parameters close to the Vela pulsar should also show pulsed emission, with the cut-off clearly at lower energies (∼50 GeV) than that observed for the Crab pulsar. However, such tail emission is not expected in pulsars with parameters close to the Geminga pulsar. The model also predicts the tail γ-ray emission extending up to ∼50 GeV from some millisecond pulsars with extreme parameters, such as PSR J0218+4243 and PSR J1823–3021A.