  We present the results of an ultra-deep radio continuum survey, containing
$\sim480$ 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 $\mu Jy~\textrm{beam}^{-1}$
at 5.5 GHz, 940 $nJy~\textrm{beam}^{-1}$ at 9 GHz, and 790
$nJy~\textrm{beam}^{-1}$ 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
($6.3\pm1.2$ $\mu Jy$ at 5.5 GHz, $5.4\pm0.9$ $\mu Jy$ at 9 GHz), flat-spectrum
($\alpha=-0.31\pm0.54$) 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 $\sim54-6000$ M$_{\odot}$, indicating an
intermediate-mass black hole or a heavy stellar-mass black hole.
