We present the results of an ultra-deep radio continuum survey, containing ~ 480 hours of observations, of the Galactic globular cluster 47 Tucanae with the Australia Telescope Compact Array. This comprehensive coverage of the cluster allows us to reach RMS noise levels of 1.19 uJY beam -1 at 5.5 GHz, 940 nJy beam-1 at 9 GHz, and 790 nJybeam-1 in a stacked 7.25 GHz image. This is the deepest radio image of a globular cluster, and the deepest image ever made with the Australia Telescope Compact Array. We identify ATCA J002405.702-720452.361, a faint (6.3+1.2uJy at 5.5 GHz, 5.4+0.9uJy at 9 GHz), flat-spectrum (a=-0.31+0.54) radio source that is positionally coincident with the cluster centre and potentially associated with a faint X-ray source. No convincing optical counterpart was identified. We use radio, X-ray, optical, and UV data to show that explanations involving a background active galactic nucleus, a chromospherically active binary, or a binary involving a white dwarf are unlikely. The most plausible explanations are that the source is an undiscovered millisecond pulsar or a weakly accreting black hole. If the X-ray source is associated with the radio source, the fundamental plane of black hole activity suggests a black hole mass of ~54-6000M⊙, indicating an intermediate-mass black hole or a heavy stellar-mass black hole
