Context. This paper presents a study of the galaxy Lyman-alpha luminosity function (LF) using a large sample of 17 lensing clusters observed by the Multi Unit Spectroscopic Explorer (MUSE) at the ESO Very Large Telescope (VLT). The magnification resulting from strong gravitational lensing by clusters of galaxies and MUSE spectroscopic capabilities allows for blind detections of LAEs without any photometric pre-selection, reaching the faint luminosity regime. Aims. The present work aims to constrain the abundance of Lyman-alpha emitters (LAEs) and quantify their contribution to the total cosmic reionization budget. Methods. We selected 600 lensed LAEs behind these clusters in the redshift range of 2.9 < z < 6.7, covering four orders of magnitude in magnification-corrected Ly-alpha luminosity (39.0 < log(L)[erg s(-1)] < 43.0). These data were collected behind lensing clusters, indicating an increased complexity in the computation of the LF to properly account for magnification and dilution effects. We applied a non-parametric V-max method to compute the LF to carefully determine the survey volume where an individual source could have been detected. The method used in this work follows the recipes originally developed in previous works, with some improvements to better account for the effects of lensing when computing the effective volume. Results. The total co-moving volume at 2.9 < z < 6.7 in the present survey is similar to 50 10(3) Mpc(3). Our LF points in the bright end (log(L) [erg s(-1)] > 42) are consistent with those obtained from blank field observations. In the faint luminosity regime, the density of sources is well described by a steep slope, alpha similar to -2 for the global redshift range. Up to log(L) [erg s(-1)] similar to 41, the steepening of the faint end slope with redshift, suggested in earlier works, is observed, but the uncertainties are still large. A significant flattening is observed towards the faintest end, for the highest redshift bins, namely, log(L)[erg s(-1)] < 41. Conclusions. When taken at face value, the steep slope at the faint-end causes the star formation rate density (SFRD) to dramatically increase with redshift, implying that LAEs could play a major role in the process of cosmic reionization. The flattening observed towards the faint end for the highest redshift bins still requires further investigation. This turnover is similar to the one observed for the UV LF at z >= 6 in lensing clusters, with the same conclusions regarding the reliability of current results. Improving the statistical significance of the sample in this low-luminosity high-redshift regime is a difficult endeavour that may lead to potentially valuable leads in understanding the process of reionization.