The MUSE Deep Lensed Field on the Hubble Frontier Field MACS~J0416

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. Methods: We describe Deep MUSE observations of 17.1 hours integration on a single pointing over the Hubble Frontier Field galaxy cluster MACS~J0416. 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. 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], [~400-3] parsec and 1<z<6.6, respectively, with the most magnified ones probing possible single gravitationally bound star clusters. The depth of the MDLF combined with lensing magnification lead us to reach a detection limit for unresolved emission lines of a few 10$^{-20}$ erg/s/cm2, after correction for lensing magnification. Ultraviolet high-ionization metal lines (and HeII1640) are detected with S/N>10 for individual objects down to de-lensed magnitude 28-30 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 clumps (down to 100-200 pc scale);(3) in some cases break down such clumps into bound star clusters (<20 pc scale);(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 that JWST and future ELTs will have when combined to study gravitational telescopes.[abridged]