Context. The Lyα emitter (LAE) fraction, XLAE, is a potentially powerful probe of the evolution of the intergalactic neutral hydrogen gas fraction. However, uncertainties in the measurement of XLAE are still under debate.Aims. Thanks to deep data obtained with the integral field spectrograph Multi Unit Spectroscopic Explorer (MUSE), we can measure the evolution of the LAE fraction homogeneously over a wide redshift range of z ≈ 3–6 for UV-faint galaxies (down to UV magnitudes of M1500 ≈ −17.75). This is a significantly fainter range than in former studies (M1500 ≤ −18.75) and it allows us to probe the bulk of the population of high-redshift star-forming galaxies.Methods. We constructed a UV-complete photometric-redshift sample following UV luminosity functions and measured the Lyα emission with MUSE using the latest (second) data release from the MUSE Hubble Ultra Deep Field Survey.Results. We derived the redshift evolution of XLAE for M1500 ∈ [ − 21.75; −17.75] for the first time with a equivalent width range EW(Lyα) ≥ 65 Å and found low values of XLAE ≲ 30% at z ≲ 6. The best-fit linear relation is XLAE = 0.07+0.06−0.03z − 0.22+0.12−0.24. For M1500 ∈ [ − 20.25; −18.75] and EW(Lyα) ≥ 25 Å, our XLAE values are consistent with those in the literature within 1σ at z ≲ 5, but our median values are systematically lower than reported values over the whole redshift range. In addition, we do not find a significant dependence of XLAE on M1500 for EW(Lyα) ≥ 50 Å at z ≈ 3–4, in contrast with previous work. The differences in XLAE mainly arise from selection biases for Lyman Break Galaxies (LBGs) in the literature: UV-faint LBGs are more easily selected if they have strong Lyα emission, hence XLAE is biased towards higher values when those samples are used.Conclusions. Our results suggest either a lower increase of XLAE towards z ≈ 6 than previously suggested, or even a turnover of XLAE at z ≈ 5.5, which may be the signature of a late or patchy reionization process. We compared our results with predictions from a cosmological galaxy evolution model. We find that a model with a bursty star formation (SF) can reproduce our observed LAE fractions much better than models where SF is a smooth function of time.Key words: dark ages / reionization / first stars / early Universe / cosmology: observations / galaxies: evolution / galaxies: high-redshift / intergalactic medium