At high metallicity, a majority of massive stars have at least one close stellar companion. The evolution of such binaries is subject to strong interaction processes, which heavily impact the characteristics of their life-ending supernova and compact remnants. For the low-metallicity environments of high-redshift galaxies, constraints on the multiplicity properties of massive stars over the separation range leading to binary interaction are crucially missing. Here we show that the presence of massive stars in close binaries is ubiquitous, even at low metallicity. Using the Very Large Telescope, we obtained multi-epoch radial velocity measurements of a representative sample of 139 massive O-type stars across the Small Magellanic Cloud, which has a metal content of about one-fifth of the solar value. We find that 45% of them show radial velocity variations that demonstrate that they are members of close binary systems, and predominantly have orbital periods shorter than 1 year. Correcting for observational biases indicates that at least 7 0−6+11% of the O stars in our sample are in close binaries, and that at least 6 8−8+7% of all O stars interact with a companion star during their lifetime. We found no evidence supporting a statistically significant trend of the multiplicity properties with metallicity. Our results indicate that multiplicity and binary interactions govern the evolution of massive stars and determine their cosmic feedback and explosive fates.