Context. Faint galaxies are theorised to have played a major role, perhaps the dominant role, in reionising the Universe. Their properties, as well as the Lyman-α emitter (LAE) fraction, X_LAE, could provide useful insights into this epoch.

Aims. We used four clusters of galaxies from the Lensed Lyman-alpha MUSE Arcs Sample (LLAMAS) that also have deep HST photometry to select a population of intrinsically faint Lyman break galaxies (LBGs) and LAEs. We study the interrelation between these two populations, their properties, and the fraction of LBGs that display Lyman-α emission.

Methods. The use of lensing clusters allows us to access an intrinsically faint population of galaxies, the largest such sample collected for this purpose: 263 LAEs and 972 LBGs with redshifts between 2.9 and 6.7, Lyman-α luminosities in the range 39.5 ≲ log(L_Lyα)(erg s⁻¹)≲42, and absolute UV magnitudes in the range −22 ≲ M₁₅₀₀ ≲ −12. In addition to matching LAEs and LBGs, we define an LAE+continuum sample for the LAEs that match with a continuum object that is not selected as an LBG. Additionally, with the use of MUSE integral field spectroscopy, we detect a population of LAEs completely undetected in the continuum.

Results. We find a redshift evolution of X_LAE in line with literature results, with diminished values above z = 6. In line with past studies, we take this as signifying an increasingly neutral intervening intergalactic medium. When inspecting this redshift evolution with different limits on EW_Lyα and M₁₅₀₀, we find that the X_LAE for the UV-brighter half of our sample is higher than the X_LAE for the UV-fainter half, a difference that increases at higher redshifts. This is a surprising result and can be interpreted as the presence of a population of low Lyman-α equivalent width (EW_Lyα), UV-bright galaxies situated in reionised bubbles and overdensities. This result is especially interesting in the context of similar, UV-bright, low EW_Lyα objects recently detected during and around the epoch of reionisation. For intrinsically fainter objects, we confirm the previously observed trend of LAEs among LBGs as galaxies with high star formation rates and low dust content, as well as the trend of the strongest LAEs having, in general, fainter M₁₅₀₀ and steeper UV slopes.