Context. The Lyα emitter (LAE) fraction, X_LAE, is a potentially powerful probe of the evolution of the intergalactic neutral hydrogen gas fraction. However, uncertainties in the measurement of X_LAE 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 X_LAE for M₁₅₀₀ ∈ [ − 21.75; −17.75] for the first time with a equivalent width range EW(Lyα) ≥ 65 Å and found low values of X_LAE ≲ 30% at z ≲ 6. The best-fit linear relation is X_LAE = 0.07^+0.06_−0.03z − 0.22^+0.12_−0.24. For M₁₅₀₀ ∈ [ − 20.25; −18.75] and EW(Lyα) ≥ 25 Å, our X_LAE 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 X_LAE on M₁₅₀₀ for EW(Lyα) ≥ 50 Å at z ≈ 3–4, in contrast with previous work. The differences in X_LAE 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 X_LAE is biased towards higher values when those samples are used. Conclusions. Our results suggest either a lower increase of X_LAE towards z ≈ 6 than previously suggested, or even a turnover of X_LAE 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.