The Binarity at LOw Metallicity (BLOeM) survey is an ESO large programme designed to obtain multi-epoch spectroscopy for 929 massive stars in the Small Magellanic Cloud (SMC). It will provide binary fractions and orbital configurations of binary systems and search for dormant black hole binary candidates (OB+BH). We present projected rotational velocities (v sin i) of all sources, and, using the multiplicity properties presented in previous papers, we derive the v sin i distributions of apparent single stars, single-lined spectroscopic (SB1) binaries, and SB2 systems. We identify a locus in the Hertzsprung-Russell diagram where rotational velocities decrease significantly; we interpret this feature as broadly corresponding to the terminal-age main sequence. The main-sequence cohort is distinguished by a broad range of v sin i values, but with a strong peak in the distribution in the range 30─60 km s−1, which is close to the resolution limit of ∼30 km s−1, indicating the presence of many upper limits. Sources in this low v sin i peak are distributed throughout the main sequence and are also present in the SB1 sample, though less prominent than in the single-star distribution. A preliminary analysis of the lowest v sin i cohort, which includes SB1 systems, implies that roughly one-third may be nitrogen rich, and we speculate that this cohort is a mix of pristine single stars, long-period binaries, and merger products. The SB2 systems appear to be mostly short-period binaries in synchronous rotation, and their v sin i estimates are distributed around a mean value of ∼140 km s−1. Higher v sin i sources are also present in the single and SB1 systems, all of which have a tail to higher v sin i values. This is consistent with tidal and mass-transfer effects. The supergiants, with a few exceptions, have low v sin i, and the bulk of these systems is essentially unresolved at the current spectral resolution (∼30 km s−1).