We present a catalogue of 2210 local (z<0.1) galaxies that contain faint active galactic nuclei (AGN). We select these objects by identifying galaxies that exhibit a significant excess in their radio luminosities, compared to what is expected from the observed levels of star formation activity in these systems. This is achieved by comparing the optical (spectroscopic) star formation rate (SFR) to the 1.4GHz luminosity measured from the Faint Images of the Radio Sky at Twenty centimeters survey. The majority of the AGN identified in this study are fainter than those in previous work, such as in the Best and Heckman (2012, Cat. J/MNRAS/421/1569) catalogue. We show that these faint AGN make a non-negligible contribution to the radio luminosity function at low luminosities (below 10^22.5^W/Hz), and host ~13 per cent of the local radio luminosity budget. Their host galaxies are predominantly high stellar-mass systems (with a median stellar mass of 1011 M), are found across a range of environments (but typically in denser environments than star-forming galaxies) and have early-type morphologies. This study demonstrates a general technique to identify AGN in galaxy populations where reliable optical SFRs can be extracted using spectro-photometry and where radio data are also available so that a radio excess can be measured. Our results also demonstrate that it is unsafe to infer SFRs from radio emission alone, even if bright AGN have been excluded from a sample, since there is a significant population of faint radio AGN that may contaminate the radio-derived SFRs.