We have used deep ground-based imaging in the near-infrared (near-IR) to search for counterparts to the luminous submillimetre (submm) sources in the catalogue of Smail et al. For the majority of the submm sources the near-IR imaging supports the counterparts originally selected from deep optical images. However, in two cases (10 per cent of the sample) we find a relatively bright near-IR source close to the submm position, sources that were unidentified in the deep Hubble Space Telescope (HST) and ground-based R-band images used by Smail et al. We place limits on colours of these sources from deep high-resolution Keck II imaging and find they have 2σ limits of (I−K)≳6.8 and (I−K)≳6.0, respectively. Both sources thus class as extremely red objects (EROs). Using the spectral properties of the submm source in the radio and submm we argue that these EROs are probably the source of the submm emission, rather than the bright spiral galaxies previously identified by Smail et al. This connection provides important insights into the nature of the enigmatic ERO population and faint submm galaxies in general. From the estimated surface density of these submm-bright EROs we suggest that this class accounts for the majority of the reddest members of the ERO population, in good agreement with the preliminary conclusions of pointed submm observations of individual EROs. We conclude that the most extreme EROs represent a population of dusty, ultraluminous galaxies at high redshifts; further study of these will provide useful insights into the nature of star formation in obscured galaxies in the early Universe. The identification of similar counterparts in blank-field submm surveys will be extremely difficult owing to their faintness (K∼20.5, I≳26.5). Finally, we discuss the radio and submm properties of the two submm-bright EROs discovered here and suggest that both galaxies lie at z≳2.