  $\omega$ Centauri is considered the most massive globular cluster of the
Milky Way and likely the former nuclear star cluster of a galaxy accreted by
the Milky Way. It is speculated to contain an intermediate-mass black hole
(IMBH) from several dynamical models. However, uncertainties regarding the
location of the cluster center or the retention of stellar remnants limit the
robustness of the IMBH detections reported so far. In this paper, we derive and
study the stellar kinematics from the highest-resolution spectroscopic data
yet, using the Multi Unit Spectroscopic Explorer (MUSE) in the narrow field
mode (NFM) and wide field mode (WFM). Our exceptional data near the center
reveal for the first time that stars within the inner 20" ($\sim$0.5 pc)
counter-rotate relative to the bulk rotation of the cluster. Using this
dataset, we measure the rotation and line-of-sight velocity dispersion (LOSVD)
profile out to 120$''$ with different centers proposed in the literature. We
find that the velocity dispersion profiles using different centers match well
with those previously published. Based on the counter--rotation, we determine a
kinematic center and look for any signs of an IMBH using the high-velocity
stars close to the center. We do not find any significant outliers $>$60 km/s
within the central 20$''$, consistent with no IMBH being present at the center
of $\omega$ Centauri. A detailed analysis of Jeans' modeling of the putative
IMBH will be presented in the next paper of the series.
