We used high-resolution spectra acquired with the Magellan Telescope to measure radial and rotational velocities of approximately 200 stars in the Galactic globular cluster NGC 3201. The surveyed sample includes blue straggler stars (BSSs) and reference stars in different evolutionary stages (main-sequence turnoff, subgiant, red giant, and asymptotic giant branches). The average radial velocity value (〈V r 〉 = 494.5 ± 0.5 km s−1) confirms a large systemic velocity for this cluster and was used to distinguish 33 residual field interlopers. The final sample of member stars has 67 BSSs and 114 reference stars. Similarly to what is found in other clusters, the totality of the reference stars has negligible rotation (< 20 km s−1), while the BSS rotational velocity distribution shows a long tail extending up to ∼200 km s−1, with 19 BSSs (out of 67) spinning faster than 40 km s−1. This sets the percentage of fast-rotating BSSs to ∼28%. Such a percentage is roughly comparable to that measured in other loose systems (ω Centauri, M4, and M55) and significantly larger than that measured in high-density clusters (as 47 Tucanae, NGC 6397, NGC 6752, and M30). This evidence supports a scenario where recent BSS formation (mainly from the evolution of binary systems) is occurring in low-density environments. We also find that the BSS rotational velocity tends to decrease for decreasing luminosity and surface temperature, similarly to what is observed in main-sequence stars. Hence, further investigations are needed to understand the impact of BSS internal structure on the observed rotational velocities.