  We present very long baseline interferometry (VLBI) observations of 179 radio
sources in the COSMOS field with extremely high sensitivity using the Green
Bank Telescope (GBT) together with the Very Long Baseline Array (VLBA)
(VLBA+GBT) at 1.4 GHz, to explore the faint radio population in the flux
density regime of tens of $\mu$Jy. Here, the identification of active galactic
nuclei (AGN) is based on the VLBI detection of the source, i.e., it is
independent of X-ray or infrared properties. The milli-arcsecond resolution
provided by the VLBI technique implies that the detected sources must be
compact and have large brightness temperatures, and therefore they are most
likely AGN (when the host galaxy is located at z$\geq$0.1). On the other hand,
this technique allows us to only positively identify when a radio-active AGN is
present, i.e., we cannot affirm that there is no AGN when the source is not
detected. For this reason, the number of identified AGN using VLBI should be
always treated as a lower limit. We present a catalogue containing the 35 radio
sources detected with the VLBA+GBT, 10 of which were not previously detected
using only the VLBA. We have constructed the radio source counts at 1.4 GHz
using the samples of the VLBA and VLBA+GBT detected sources of the COSMOS field
to determine a lower limit for the AGN contribution to the faint radio source
population. We found an AGN contribution of >40-75% at flux density levels
between 150 $\mu$Jy and 1 mJy. This flux density range is characterised by the
upturn of the Euclidean-normalised radio source counts, which implies a
contribution of a new population. This result supports the idea that the
sub-mJy radio population is composed of a significant fraction of
radio-emitting AGN, rather than solely by star-forming galaxies, in agreement
with previous studies.
