We present new and accurate radial velocity (RV) measurements of luminous stars of all ages (old horizontal branch, intermediate-age red clump, and young blue plume, as well as red giants of a range of ages: 20.6 <= V <= 22) in the Carina dwarf spheroidal galaxy, based on low-resolution spectra collected with the FORS2 multiobject slit spectrograph at the VLT. This data set was complemented by RV measurements based on medium- and high-resolution archive spectra of brighter (V less than or similar to 20.6) Carina targets collected with the GIRAFFE multiobject fiber spectrograph at the VLT. The combined sample includes more than 21,340 individual spectra of approximate to 2000 stars covering the entire body of the galaxy. The mean (< RV > = 220.4 +/- 0.1 km s(-1)) and the dispersion (sigma = 11.7 +/- 0.1 km s(-1)) of the RV distribution of candidate Carina stars (similar to 1210 objects, 180 <= RV <= 260 km s(-1), 4 sigma) agree quite well with similar measurements available in the literature. To further improve the statistics, the accurate RV measurements recently provided by Walker et al. were also added to the current data set. We ended up with a sample of similar to 1370 RV measurements of candidate Carina stars that is approximate to 75% larger than any previous Carina RV sample. We found that the hypothesis that the Carina RV distribution is Gaussian can be discarded at 99% confidence level. The mean RV across the body of the galaxy varies from similar to 220 km s(-1) at a distance of 7' (similar to 200 pc) from the center to similar to 223 km s(-1) at 13' (similar to 400 pc, 6 sigma level) and flattens out to a constant value of similar to 221 km s(-1) at larger distances (600 pc, 4 sigma level). Moreover, and even more importantly, we found that in the Carina regions where the mean RV is smaller, the dispersion is also smaller, and the RV distribution is more centrally peaked (i.e., the kurtosis attains larger values). The difference in mean RV is more than 4 km s(-1) (9 sigma level) when moving from east to west and more than 3 km s(-1) (similar to 7 sigma level) when moving from north to south. The RV gradient appears to be in the direction of the Carina proper motion. However, this parameter is affected by large uncertainties to reach a firm conclusion. There is evidence of a secondary maximum in RV across the Carina center (vertical bar D vertical bar <= 200 pc). The occurrence of a secondary feature across the Carina center is also supported by the flat-topped radial distribution based on the photometric catalog. These findings are reminiscent of a substructure with transition properties that are already detected in dwarf ellipticals, and they call for confirmation by independent investigations.