  Background. Astrometry at or below the micro-arcsec level with an imaging
telescope assumes that the uncertainty on the location of an unresolved source
can be an arbitrarily small fraction of the detector pixel, given a sufficient
photon budget. Aim. This paper investigates the geometric limiting precision,
in terms of CCD pixel fraction, achieved by a large set of star field images,
selected among the publicly available science data of the TESS mission. Method.
The statistics of the distance between selected bright stars ($G \simeq
5\,mag$), in pixel units, is evaluated, using the position estimate provided in
the TESS light curve files. Results. The dispersion of coordinate differences
appears to be affected by long term variation and noisy periods, at the level
of $0.01$ pixel. The residuals with respect to low-pass filtered data (tracing
the secular evolution), which are interpreted as the experimental astrometric
noise, reach the level of a few milli-pixel or below, down to $1/5,900$ pixel.
Saturated images are present, evidencing that the astrometric precision is
mostly preserved across the CCD columns, whereas it features a graceful
degradation in the along column direction. The cumulative performance of the
image set is a few micro-pixel across columns, or a few 10 micro-pixel along
columns. Conclusions. The idea of astrometric precision down to a small
fraction of a CCD pixel, given sufficient signal to noise ratio, is confirmed
by real data from an in-flight science instrument to the $10^{-6}$ pixel level.
Implications for future high precision astrometry missions are briefly
discussed.
