Scattering of Lyα photons by neutral hydrogen gas in a single outflowing ‘supershell’ around star-forming regions often explains the shape and offset of the observed Lyα emission line from galaxies. We compute the radiation pressure that is exerted by this scattered Lyα radiation on the outflowing material. We show that for reasonable physical parameters, Lyα radiation pressure alone can accelerate supershells to velocities in the range vsh= 200–400 km s−1. These supershells possibly escape from the gravitational potential well of their host galaxies and contribute to the enrichment of the intergalactic medium. We compute the physical properties of expanding supershells that are likely to be present in a sample of known high-redshift (z= 2.7–5.0) galaxies, under the assumption that they are driven predominantly by Lyα radiation pressure. We predict ranges of radii rsh= 0.1–10 kpc, ages tsh= 1–100 Myr and energies Esh= 1053–1055 erg, which are in reasonable agreement with the properties of local galactic supershells. Furthermore, we find that the radius, rsh, of a Lyα-driven supershell of constant mass depends uniquely on the intrinsic Lyα luminosity of the galaxy, Lα, the H i column density of the supershell, NH I, and the shell speed, vsh, through the scaling relation rsh∝Lα/(NH Iv2sh). We derive mass outflow rates in supershells that reach ∼10–100 per cent of the star formation rates of their host galaxies.