Context. The blue supergiant (BSG) domain contains a large variety of stars whose past and future evolutionary paths are still highly uncertain. Since binary interaction plays a crucial role in the fate of massive stars, investigating the multiplicity among BSGs helps shed light on the fate of such objects. Aims. We aim to estimate the binary fraction of a large sample of BSGs in the Small Magellanic Cloud (SMC) within the Binarity at LOw Metallicity (BLOeM) survey. In total, we selected 262 targets with spectral types B0-B3 and luminosity classes I-II. Methods. This work is based on spectroscopic data collected by the FLAMES instrument, mounted on the Very Large Telescope, which gathered nine epochs over three months. Our spectroscopic analysis for each target includes the individual and peak-to-peak radial velocity measurements, an investigation of the line profile variability, and a periodogram analysis to search for possible short- and long-period binaries. Results. By applying a 20 km s‑1 threshold on the peak-to-peak radial velocities above which we would consider the star to be binary, the resulting observed spectroscopic binary fraction for our BSG sample is 23 ± 3%. An independent analysis of line profile variability reveals 11 (plus 5 candidates) double-lined spectroscopic binaries and 32 (plus 41 candidates) single-lined spectroscopic binaries. Based on these results, we estimated the overall observed binary fraction in this sample to be 34 ± 3%, which is close to the computed intrinsic binary fraction of 40 ± 4%. In addition, we derived reliable orbital periods for 41 spectroscopic binaries and potential binary candidates, among which there are 17 eclipsing binaries, including 20 SB1 and SB2 systems with periods of less than 10 days. We reported a significant drop in the binary fraction of BSGs with spectral types later than B2 and effective temperatures less than 18 kK, which could indicate the end of the main sequence phase in this temperature regime. We found no metallicity dependence in the binary fraction of BSGs, compared to existing spectroscopic surveys of the Galaxy and Large Magellanic Cloud.