We present a comprehensive analysis of the presence of very massive stars (VMS > 100 M⊙) in the integrated spectra of 13 UV-bright star-forming galaxies at 2.2 ≲ z ≲ 3.6 taken with the Gran Telescopio Canarias (GTC). These galaxies have very high UV absolute magnitudes (MUV ≃ −24), intense star formation (star formation rate ≃100 − 1000 M⊙ yr−1), and metallicities in the range of 12 + log(O/H) ≃ 8.10 − 8.50 inferred from strong rest-optical lines. The GTC rest-UV spectra reveal spectral features indicative of very young stellar populations with VMS, such as strong P-Cygni line profiles in the wind lines N Vλ1240 and C IVλ1550 along with intense and broad He IIλ1640 emission with equivalent width (EW0) ≃ 1.40 − 4.60 Å, and full width half maximum (FWHM) ≃1150 − 3170 km s−1. A Comparison with known VMS-dominated sources and typical galaxies without VMS reveals that some UV-bright galaxies closely resemble VMS-dominated clusters (e.g., R136 cluster). The presence of VMS is further supported by a quantitative comparison of the observed strength of the He II emission with population synthesis models with and without VMS, where models with VMS are clearly preferred. Employing an empirical threshold for EW0 (He II) ≥ 3.0 Å, along with the detection of other VMS-related spectral profiles (N IVλ1486, 1719), we classify nine out of 13 UV-bright galaxies as VMS-dominated sources. This high incidence of VMS-dominated sources in the UV-bright galaxy population (≈70%) contrasts significantly with the negligible presence of VMS in typical LUV∗ LBGs at similar redshifts (< 1%). Our results thus indicate that VMS are common in UV-bright galaxies, suggesting a different initial mass function (IMF) with upper mass limits between 175 M⊙ and 475 M⊙.Key words: stars: massive / galaxies: high-redshift / galaxies: starburst / ultraviolet: galaxies⋆ A copy of the reduced spectra is available at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/686/A185