Young massive clusters play an important role in the evolution of their host galaxies, and feedback from the high-mass stars in these clusters can have profound effects on the surrounding interstellar medium. The nuclear starburst in the nearby galaxy NGC253 at a distance of 3.5 Mpc is a key laboratory in which to study star formation in an extreme environment. Previous high resolution (1.9 pc) dust continuum observations from ALMA discovered 14 compact, massive super star clusters (SSCs) still in formation. We present here ALMA data at 350 GHz with 28 milliarcsecond (0.5 pc) resolution. We detect blueshifted absorption and redshifted emission (P-Cygni profiles) towards three of these SSCs in multiple lines, including CS 7$-$6 and H$^{13}$CN 4$-$3, which represents direct evidence for previously unobserved outflows. The mass contained in these outflows is a significant fraction of the cluster gas masses, which suggests we are witnessing a short but important phase. Further evidence of this is the finding of a molecular shell around the only SSC visible at near-IR wavelengths. We model the P-Cygni line profiles to constrain the outflow geometry, finding that the outflows must be nearly spherical. Through a comparison of the outflow properties with predictions from simulations, we find that none of the available mechanisms completely explains the observations, although dust-reprocessed radiation pressure and O star stellar winds are the most likely candidates. The observed outflows will have a very substantial effect on the clusters evolution and star formation efficiency.