© 2019 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society Detections and non-detections of Lyman alpha (Lyα) emission from z > 6 galaxies (<1 Gyr after the big bang) can be used to measure the timeline of cosmic reionization. Of key interest to measuring reionization's mid-stages, but also increasing observational challenge, are observations at z > 7, where Lyα redshifts to near infra-red wavelengths. Here we present a search for z > 7.2 Lyα emission in 53 intrinsically faint Lyman Break Galaxy candidates, gravitationally lensed by massive galaxy clusters, in the KMOS Lens-Amplified Spectroscopic Survey (KLASS). With integration times of ∼7-10 h, we detect no Lyα emission with signal-to-noise ratio (S/N) > 5 in our sample. We determine our observations to be 80 per cent complete for 5σ spatially and spectrally unresolved emission lines with integrated line flux >5.7 × 10−18 erg s−1 cm−2. We define a photometrically selected sub-sample of 29 targets at z = 7.9 ± 0.6, with a median 5σ Lyα EW limit of 58 Å. We perform a Bayesian inference of the average intergalactic medium (IGM) neutral hydrogen fraction using their spectra. Our inference accounts for the wavelength sensitivity and incomplete redshift coverage of our observations, and the photometric redshift probability distribution of each target. These observations, combined with samples from the literature, enable us to place a lower limit on the average IGM neutral hydrogen fraction of > 0.76 (68 per cent), > 0.46 (95 per cent) at z ∼ 8, providing further evidence of rapid reionization at z ∼ 6-8. We show that this is consistent with reionization history models extending the galaxy luminosity function to MUV −12, with low ionizing photon escape fractions, fesc 15 per cent.