  Context. Gamma Doradus (hereafter $\gamma$~Dor) stars are gravity-mode
pulsators whose periods carry information about the internal structure of the
star. These periods are especially sensitive to the internal rotation and
chemical mixing, two processes that are currently not well constrained in the
theory of stellar evolution. Aims. We aim to identify the pulsation modes and
deduce the internal rotation and buoyancy travel time for 106 $\gamma$ Dor
stars observed by the Transiting Exoplanet Survey Satellite (TESS) mission in
its southern continuous viewing zone (hereafter S-CVZ). We rely on 140
previously detected period-spacing patterns, that is, series of
(near-)consecutive pulsation mode periods. Methods. We used the asymptotic
expression to compute gravity-mode frequencies for ranges of the rotation rate
and buoyancy travel time that cover the physical range in $\gamma$~Dor stars.
Those frequencies were fitted to the observed period-spacing patterns by
minimising a custom cost function. The effects of rotation were evaluated using
the traditional approximation of rotation, using the stellar pulsation code
GYRE. Results. We obtained the pulsation mode identification, internal rotation
and buoyancy travel time for 60 TESS $\gamma$~Dor stars. For the remaining 46
targets, the detected patterns are either too short or contained too many
missing modes for unambiguous mode identification, and longer light curves are
required. For the successfully analysed stars, we found that period-spacing
patterns from 1-yr long TESS light curves can constrain the internal rotation
and buoyancy travel time to a precision of $\rm 0.03~d^{-1}$ and 400s,
respectively, which is about half as precise as literature results based on
4-yr Kepler light curves of $\gamma$~Dor stars.
