We present the discovery of compact, obscured star formation in galaxies at z ~ 0.6 that exhibit 1000 km s–1 outflows. Using optical morphologies from the Hubble Space Telescope and infrared photometry from the Wide-field Infrared Survey Explorer, we estimate star formation rate (SFR) surface densities that approach ΣSFR ≈ 3000 M ☉ yr–1 kpc–2, comparable to the Eddington limit from radiation pressure on dust grains. We argue that feedback associated with a compact starburst in the form of radiation pressure from massive stars and ram pressure from supernovae and stellar winds is sufficient to produce the high-velocity outflows we observe, without the need to invoke feedback from an active galactic nucleus