We report the discovery of a high-redshift, massive molecular outflow in the starburst galaxy SPT 0346-52 (z=5.656) via the detected absorption of high-excitation water transitions (H_2_O 4_2,3_-4_1,4_ and H_2_O 3_3,0_-3_2,1_) with the Atacama Large Millimeter/submillimeter Array (ALMA). The host galaxy is one of the most powerful starburst galaxies at high redshift (star formation rate; SFR~3600M_{sun}_/yr), with an extremely compact (~320pc) star formation region and a SFR surface density ({SIGMA}_SFR_~5500M_{sun}_/yr/kpc^2^) five times higher than "maximum" (i.e. Eddington-limited) starbursts, implying a highly transient phase. The estimated outflow rate is ~500M_{sun}_/yr, which is much lower than the SFR, implying that in this extreme starburst the outflow capabilities saturate and the outflow is no longer capable of regulating star formation, resulting in a runaway process in which star formation will use up all available gas in less than 30Myr. Finally, while previous kinematic investigations of this source revealed possible evidence for an ongoing major merger, the coincidence of the hyper-compact starburst and high-excitation water absorption indicates that this is a single starburst galaxy surrounded by a disc.