We present the results of an extensive survey for rapid pulsators among Extreme Horizontal Branch (EHB) stars in ω Cen. The observations performed consist of nearly 100 h of time-series photometry for several off-centre fields of the cluster, as well as low-resolution spectroscopy for a partially overlapping sample. We obtained photometry for some 300 EHB stars, for around half of which we are able to recover light curves of sufficient quality to either detect or place meaningful non-detection limits for rapid pulsations. Based on the spectroscopy, we derive reliable values of log g, Teff and log N(He) /N(H) for 38 targets, as well as good estimates of the effective temperature for another nine targets, whose spectra are slightly polluted by a close neighbour in the image. The survey uncovered a total of five rapid variables with multi-periodic oscillations between 85 and 125 s. Spectroscopically, they form a homogeneous group of hydrogen-rich subdwarf O stars clustered between 48 000 and 54 000 K. For each of the variables we are able to measure between two and three significant pulsations believed to constitute independent harmonic oscillations. However, the interpretation of the Fourier spectra is not straightforward due to significant fine structure attributed to strong amplitude variations. In addition to the rapid variables, we found an EHB star with an apparently periodic luminosity variation of ~2700 s, which we tentatively suggest may be caused by ellipsoidal variations in a close binary. Using the overlapping photometry and spectroscopy sample we are able to map an empirical ω Cen instability strip in log g - Teff space. This can be directly compared to the pulsation driving predicted from the Montréal "second-generation" models regularly used to interpret the pulsations in hot B subdwarfs. Extending the parameter range of these models to higher temperatures, we find that the region where p-mode excitation occurs is in fact bifurcated, and the well-known instability strip between 29 000-36 000 K where the rapid subdwarf B pulsators are found is complemented by a second one above 50 000 K in the models. While significant challenges remain at the quantitative level, we believe that the same κ-mechanism that drives the pulsations in hot B subdwarfs is also responsible for the excitation of the rapid oscillations observed in the ω Cen variables. Intriguingly, the ω Cen variables appear to form a unique class. No direct counterparts have so far been found either in the Galactic field, nor in other globular clusters, despite dedicated searches. Conversely, our survey revealed no ω Cen representatives of the rapidly pulsating hot B subdwarfs found among the field population, though their presence cannot be excluded from the limited sample. Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile (proposal IDs 083.D-0833, 386.D-0669, 087.D-0216 and 091.D-0791).The reduced spectra are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/589/A1