Context. A census of faint and tiny star forming complexes at high redshift is key to improving our understanding of reionizing sources, galaxy growth, and the formation of globular clusters.Aims. We present the MUSE Deep Lensed Field (MDLF) program, which is aimed at unveiling the very faint population of high redshift sources that are magnified by strong gravitational lensing and to significantly increase the number of constraints for the lens model.Methods. We describe Deep MUSE observations of 17.1 h of integration on a single pointing over the Hubble Frontier Field galaxy cluster MACS J0416, providing line flux limits down to 2x10(-19) erg s(-1) cm(-2) within 300 km s(-1) and continuum detection down to magnitude 26, both at the three sigma level at lambda =7000 angstrom. For point sources with a magnification (mu) greater than 2.5 (7.7), the MLDF depth is equivalent to integrating more than 100 (1000) h in blank fields, as well as complementing non-lensed studies of very faint high-z sources. The source-plane effective area of the MDLF with mu> 6.3 is < 50% of the image-plane field of view.Results. We confirm spectroscopic redshifts for all 136 multiple images of 48 source galaxies at 0.9< z< 6.2. Within those galaxies, we securely identify 182 multiple images of 66 galaxy components that we use to constrain our lens model. This makes MACS J0416 the cluster with the largest number of confirmed constraints for any strong lens model to date. We identify 116 clumps belonging to background high-z galaxies; the majority of them are multiple images and span magnitude, size, and redshift intervals of [-18, -10], [similar to 400-3] parsec and 1< z< 6.6, respectively, with the faintest or most magnified ones probing possible single gravitationally bound star clusters. The multiplicity introduced by gravitational lensing allows us, in several cases, to triple the effective integration time up to similar to 51 h exposure per single family, leading to a detection limit for unresolved emission lines of a few 10(-20) erg s(-1) cm(-2), after correction for lensing magnification. Ultraviolet high-ionization metal lines (and HeII lambda 1640) are detected with S/N> 10 for individual objects down to de-lensed magnitudes between 28 and 30. The median stacked spectrum of 33 sources with a median M-UV similar or equal to -17 and z = 3.2 (1.7< z< 3.9) shows high-ionization lines, suggesting that they are common in such faint sources.Conclusions. Deep MUSE observations, in combination with existing HST imaging, allowed us to: (1) confirm redshifts for extremely faint high-z sources; (2) peer into their internal structure to unveil clumps down to 100-200 pc scale; (3) in some cases, break down such clumps into star-forming complexes matching the scales of bound star clusters (< 20 pc effective radius); (4) double the number of constraints for the lens model, reaching an unprecedented set of 182 bona-fide multiple images and confirming up to 213 galaxy cluster members. These results demonstrate the power of JWST and future adaptive optics facilities mounted on the Extremely Large Telescopes (e.g., European-ELT Multi-conjugate Adaptive Optics RelaY, MAORY, coupled with the Multi-AO Imaging CamerA for Deep Observations, MICADO) or Very Large Telescope (e.g., MCAO Assisted Visible Imager and Spectrograph, MAVIS) when combined in studies with gravitational telescopes.