Scattering of Ly \alpha photons by neutral hydrogen gas in a single outflowing ’ supershell ’ around star forming regions often explains the shape and offset of the observed Ly \alpha emission line from galaxies . We compute the radiation pressure that is exerted by this scattered Ly \alpha radiation on the outflowing material . We show that for reasonable physical parameters , Ly \alpha radiation pressure alone can accelerate supershells to velocities in the range v _ { sh } = 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 Ly \alpha radiation pressure . We predict ranges of radii r _ { sh } = 0.1 – 10 kpc , ages t _ { sh } = 1 – 100 Myr , and energies E _ { sh } = 10 ^ { 53 } – 10 ^ { 55 } ergs , which are in reasonable agreement with the properties of local galactic supershells . Furthermore , we find that the radius , r _ { sh } , of a Ly \alpha -driven supershell of constant mass depends uniquely on the intrinsic Ly \alpha luminosity of the galaxy , L _ { \alpha } , the HI column density of the supershell , N _ { HI } , and the shell speed , v _ { sh } , through the scaling relation r _ { sh } \propto L _ { \alpha } / ( N _ { HI } v _ { sh } ^ { 2 } ) . We derive mass outflow rates in supershells that reach \sim 10 – 100 \% of the star formation rates of their host galaxies .