Stellar multiplicity is believed to influence planetary formation and evolution, although the precise nature and extent of this role remain ambiguous. We present a study aimed at testing the role of stellar multiplicity in the formation and/or evolution of the most massive, close-in planetary and substellar companions... Using direct imaging observations, as well as the Gaia DR2 catalogue, we searched for wide binary companions to 38 stars hosting massive giant planets or brown dwarfs (M > 7 MJup) on orbits shorter than ~1 AU. We report the discovery of a new component in the WASP-14 system, and present an independent confirmation of a comoving companion to WASP-18. From a robust Bayesian statistical analysis, we derived a binary fraction of 79.0+13.2-14.7% between 20-10,000 AU for our sample, twice as high as for field stars with a 3-{\sigma} significance. This binary frequency was found to be larger than for lower-mass planets on similar orbits, and we observed a marginally higher binary rate for inner companions with periods shorter than 10 days. These results demonstrate that stellar companions greatly influence the formation and/or evolution of these systems, suggesting that the role played by binary companions becomes more important for higher-mass planets, and that this trend may be enhanced for systems with tighter orbits. Our analysis also revealed a peak in binary separation at 250 AU, highlighting a shortfall of close binaries among our sample. This indicates that the mechanisms affecting planet and brown dwarf formation or evolution in binaries must operate from wide separations, although we found that the Kozai-Lidov mechanism is unlikely to be the dominant underlying process. We conclude that binarity plays a crucial role in the existence of very massive short-period giant planets and brown dwarf desert inhabitants, which are almost exclusively observed in multiple systems. (read more)