Synchrotron radiation of ultrarelativistic particles accelerated in a pulsar wind nebula may dominate its spectrum up to γ-ray energies. Because of the short cooling time of the γ-ray-emitting electrons and positrons, the γ-ray-emission zone is in the immediate vicinity of the acceleration site. Particle acceleration likely occurs at the termination shock of a relativistic striped wind, where multiple forced magnetic field reconnections provide strong magnetic fluctuations facilitating Fermi acceleration processes. The acceleration mechanisms imply the presence of stochastic magnetic fields in the particle acceleration region, which cause stochastic variability of the synchrotron emission. This variability is particularly strong in the steep γ-ray tail of the spectrum, where modest fluctuations of the magnetic field lead to strong flares of spectral flux. In particular, stochastic variations of magnetic field, which may lead to quasi-cyclic γ-ray flares, can be produced by the relativistic cyclotron ion instability at the termination shock. Our model calculations of the spectral and temporal evolution of synchrotron emission in the spectral cut-off regime demonstrate that the intermittent magnetic field concentrations dominate the γ-ray emission from highest energy electrons and provide fast, strong variability even for a quasi-steady distribution of radiating particles. The simulated light curves and spectra can explain the very strong γ-ray flares observed in the Crab Nebula and lack of strong variations at other wavelengths. The model predicts high polarization in the flare phase, which can be tested with future polarimetry observations.