Recent James Webb Space Telescope (JWST) observations have revealed a surprisingly abundant population of faint, dusty active galactic nuclei (AGNs) at z~4-7. Together with the presence of supermassive black holes (SMBHs) at z>6, this raises questions about the formation and growth histories of early black holes. Current theories for the formation of seed black holes from the death of the first stars (i.e. light seeds) and/or the direct collapse of primordial gas clouds (i.e. heavy seeds) still lack observational confirmation. Here, we present LID-568, a low-mass (7.2e6Msun) black hole hosting powerful outflows that is observed in an extreme phase of rapid growth at z~4. This object is similar to other JWST-discovered faint AGN populations, but is bright in X-ray emission and accreting at more than 4000% of the limit at which radiation pressure exceeds the force of gravitational attraction of the black hole (i.e. super-Eddington accretion). Analysis of JWST NIRSpec/IFU data reveals spatially extended Ha emission with velocities of ~ -600 - -500 km/s relative to the central black hole, indicative of robust nuclear-driven outflows. LID-568 represents an elusive low-mass black hole experiencing super-Eddington accretion as invoked by models of early black hole formation. This discovery showcases a previously undiscovered key parameter space and offers crucial insights into rapid black hole growth mechanisms in the early universe.