Gamma-ray binaries are massive stars with compact object companions that are observed to emit most of their energy in the gamma-ray range. One of these binaries is known to contain a radio pulsar, PSR B1259-63. Synchrotron and inverse-Compton emission from particles accelerated beyond the light cylinder in striped pulsar winds has been proposed to explain the X-ray to high-energy (HE; >100 MeV) gamma-ray emission from isolated pulsars. This pulsar model extends naturally to binary environments, where seed photons for inverse-Compton scattering are provided by the companion star. Here, we investigate the possibility of gamma-ray emission from PSR B1259-63 in the framework of the striped pulsar wind model. The orbital geometry of PSR B1259-63 is well constrained by observations, and the double radio pulse suggests an almost orthogonal rotator so that the solid angle covered by the striped region is close to 4π. We calculate the orbital and rotational phase-resolved spectral variability and light curves to expect. We find that the recent detection by the Fermi/Large Area Telescope of PSR B1259-63 close to periastron can be explained by a striped wind if the emission arises from a large range of radii (≥1000r_L). We constrain the particle density number at the light cylinder n_L≈ 7 × 10¹⁵ m^-3. The re-brightening a month after periastron passage could be due to interaction with additional seed photons from the trailing pulsar wind nebula. Striped winds may also be at work in the gamma-ray binaries LS I +61°303 and LS 5039, both of which have HE gamma-ray spectra reminiscent of those of pulsars and fluxes modulated on the orbital period. Gamma-ray pulsations are expected. Some gamma-ray binaries should be faint in HE gamma-rays (HESS J0632+057) because the line of sight does not cross the striped wind region.