We present spectroscopic confirmation of 10 highly luminous (L≥ 2L ★) Lyα emitters (LAEs) in the redshift range 6.01 < z < 6.49 (nine galaxies and one active galactic nucleus), initially drawn from a sample of 14 z phot≥ 6 Lyman-break galaxies (LBGs) selected from an area of 0.25 deg2 within the UK Infrared Deep Sky Survey (UKIDSS) Ultra-Deep Survey (UDS). Overall, our high rate of spectroscopic confirmation (≥71 per cent) and low rate of contamination provides a strong vindication of the photometric redshift analysis used to define the original sample. By considering star formation rate estimates based on the Lyα and ultraviolet continuum luminosity we conclude that our sample is consistent with a Lyα escape fraction of ≃25 per cent. Moreover, after careful consideration of the potential uncertainties and biases, we find that 40-50 per cent of our sample of L≥ 2L ★ galaxies at 6.0 < z < 6.5 display strong Lyα emission (rest-frame equivalent width ≥25 Å), a fraction which is a factor of ≃2 higher than previously reported for L≤L ★ galaxies at z≃ 6. Our results suggest that, as the epoch of reionization is approached, it is plausible that the LAE fraction amongst luminous (L≥ 2L ★) LBGs shows a similarly sharp increase to that observed in their lower luminosity (L≤L ★) counterparts.