[ABRIDGED] $omega$ Centauri (NGC 5139) contains large numbers of variable stars of different types and, in particular, more than a hundred RR Lyrae stars. We have conducted a variability survey of $omega$ Cen in the NIR, using ESOs 4.1m Visible and Infrared Survey Telescope for Astronomy (VISTA). This is the first paper of a series describing our results. $omega$ Cen was observed using VIRCAM mounted on VISTA. A total of 42 and 100 epochs in $J$ and $K_{rm S}$, respectively, were obtained, distributed over a total timespan of 352 days. PSF photometry was performed, and periods of the known variable stars were improved when necessary using an ANOVA analysis. An unprecedented homogeneous and complete NIR catalogue of RR Lyrae stars in the field of $omega$ Cen was collected, allowing us to study, for the first time, all the RR Lyrae stars associated to the cluster, except 4 located far away from the cluster center. Membership status, subclassifications between RRab and RRc subtypes, periods, amplitudes, and mean magnitudes were derived for all the stars in our sample. Additionally, 4 new RR Lyrae stars were discovered, 2 of them with high probability of being cluster members. The distribution of $omega$ Cen stars in the Bailey (period-amplitude) diagram is also discussed. Reference lines in this plane, for both Oosterhoff type I (OoI) and II (OoII) components, are provided. In the present paper, we clarify the status of many (candidate) RR Lyrae stars that had been unclear in previous studies. This includes stars with anomalous positions in the color-magnitude diagram, uncertain periods or/and variability types, and possible field interlopers. We conclude that $omega$ Cen hosts a total of 88 RRab and 101 RRc stars, for a grand total of 189 likely members. We confirm that most RRab stars in the cluster belong to an OoII component, as previously found using visual data.
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