We report the discovery of a double-peaked Lyman α profile in a galaxy at z = 6.803, A370p_z1, in the parallel Frontier Field of Abell 370. The velocity separation between the blue and red peaks of the Lyman α profile ( $\Delta v=101_{-19}^{+38} (\pm 48)\, \rm km\, \rm s^{-1}$ ) suggests an extremely high escape fraction of ionizing photons $ 59(51){{\ \rm per\ cent}} (2\sigma)$ . The spectral energy distribution indicates a young (50 Myr), star-forming ( $12\pm 6 \, \rm {M}_\odot \rm {yr}^{-1}$ ) galaxy with an IRAC excess implying strong [O III] + H β emission. On the basis of the high escape fraction measured, we demonstrate that A370p_z1 was solely capable of creating an ionized bubble sufficiently large to account for the blue component of its Lyman α profile. We discuss whether A370p_z1 may be representative of a larger population of luminous z ≃ 7 double-peaked Lyman α emitting sources with high escape fractions that self-ionized their surroundings without contributions from associated ultraviolet-fainter sources.