We discovered a strongly lensed (mu greater than or similar to 40) Ly alpha emission at z = 6.629 (S/N similar or equal to 18) in the MUSE Deep Lensed Field (MDLF) targeting the Hubble Frontier Field (HFF) galaxy cluster MACS J0416. Dedicated lensing simulations imply that the Ly alpha emitting region necessarily crosses the caustic. The arc-like shape of the Ly alpha extends 3 arcsec on the observed plane and is the result of two merged multiple images, each one with a de-lensed Ly alpha luminosity L less than or similar to 2.8 x 10(40) erg s(-1) arising from a confined region (less than or similar to 150 pc effective radius). A spatially unresolved Hubble Space Telescope (HST) counterpart is barely detected at S/N similar or equal to 2 after stacking the near-infrared bands, corresponding to an observed (intrinsic) magnitude m(1500) greater than or similar to 30.8 (greater than or similar to 35.0). The inferred rest-frame Ly alpha equivalent width is EW0 > 1120 angstrom if the IGM transmission is TIGM < 0.5. The low luminosities and the extremely large Ly a EW0 match the case of a Population III (Pop III) star complex made of several dozens stars (similar to 10(4) M-circle dot) that irradiate an HII region crossing the caustic. While the Ly alpha and stellar continuum are among the faintest ever observed at this redshift, the continuum and the Ly alpha emissions could be affected by differential magnification, possibly biasing the EW0 estimate. The aforementioned tentative HST detection tends to favour a large EW0, making such a faint Pop III candidate a key target for the James Webb Space Telescope and Extremely Large Telescopes.