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The fraction of Lyman-α emitters (LAEs) among the galaxy population has been found to increase from z ~ 0 to z ~ 6 and drop dramatically at z> 6. This drop has been interpreted as an effect of an increasingly neutral intergalactic medium (IGM) with increasing redshift, while a Lyman continuum escape fraction evolving with redshift and/or a sudden change of galaxy physical properties can also contribute to the decreasing LAE fraction. We report the result of a large VLT/FORS2 program aiming to confirm spectroscopically a large galaxy sample at z ≥ 6 that has been selected in several independent fields through the Lyman break technique. Combining those data with archival data, we create a large and homogeneous sample of z ~ 6 galaxies (N = 127), complete in terms of Lyα detection at > 95% for Lyα equivalent width EW(Lyα) ≥ 25 Å. We use this sample to derive a new measurement of the LAE fraction at z ~ 6 and derive the physical properties of these galaxies through spectral energy distribution (SED) fitting. We find a median LAE fraction at z ~ 6 lower than in previous studies, while our sample exhibits typical properties for z ~ 6 galaxies in terms of UV luminosity and UVβ slope. The comparison of galaxy physical properties between LAEs and non-LAEs is comparable to results at lower redshift: LAEs with the largest EW(Lyα) exhibit bluer UV slopes, are slightly less massive and less star-forming. The main difference between LAEs and non-LAEs is that the latter are significantly dustier. Using predictions of our SED fitting code accounting for nebular emission, we find an effective Lyα escape fraction fesceff(Lyα) = 0.23-0.17+0.36 remarkably consistent with the value derived by comparing UV luminosity function with Lyα luminosity function. We conclude that the drop in the LAE fraction from z ~ 6 to z> 6 is less dramatic than previously found and the effect of an increasing IGM neutral fraction is possibly observed at 5 <z< 6. The processes driving the escape of Lyα photons at z ~ 6 are similar to those at lower redshifts and based on our derived fesceff(Lyα), we find that the IGM has a relatively small impact on Lyα photon visibility at z ~ 6, with a lower limit for the IGM transmission to Lyα photons, TIGM ≳ 0.20, likely due to the presence of outflows