We report the identification of an extreme proto-cluster of galaxies in the early Universe whose core (nicknamed Distant Red Core, DRC) is formed by at least ten dusty star-forming galaxies (DSFGs), confirmed to lie at $z_{rm spec} = 4.002$ via detection of [CI](1-0), $^{12}$CO(6-5), $^{12}$CO(4-3), $^{12}$CO(2-1) and ${rm H_2O} (2_{11} - 2_{02})$ emission lines, detected using ALMA and ATCA. The spectroscopically-confirmed components of the proto-cluster are distributed over a ${rm 260, kpc times 310, kpc}$ region and have a collective obscured star-formation rate (SFR) of $sim 6500 , M_odot , {rm yr}^{-1}$, considerably higher than has been seen before in any proto-cluster of galaxies or over-densities of DSFGs at $z gtrsim 4$. Most of the star formation is taking place in luminous DSFGs since no Ly$alpha$ emitters are detected in the proto-cluster core, apart from a Ly$alpha$ blob located next to one of the DRC dusty components and extending over $60,{rm kpc}$. The total obscured SFR of the proto-cluster could rise to ${rm SFR} sim 14,400 , M_odot , {rm yr}^{-1}$ if all the members of an over-density of bright DSFGs discovered around DRC in a wide-field LABOCA 870-$mu$m image are part of the same structure. The total halo mass of DRC could be as high as $sim 4.4 times 10^{13},M_odot$ and could be the progenitor of a Coma-like cluster at $z = 0$. The relatively short gas-depletion times of the DRC components suggest either the presence of a mechanism able to trigger extreme star formation simultaneously in galaxies spread over a few hundred kpc or the presence of gas flows from the cosmic web able to sustain star formation over several hundred million years.
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