Context. Large-scale surveys over the last years have revealed about 300 quasi-stellar objects (QSOs) at redshifts above 6. Follow-up observations have identified surprising properties, such as the very high black hole (BH) masses, spatial correlations with surrounding cold gas of the host galaxy, and high CIV-MgII Velocity shifts. In particular, the discovery of luminous high-redshift quasars suggests that at least some BHs likely have high masses at birth and grow efficiently. Aims. Our aim is to quantify quasar pairs at high redshift for a large sample of objects. This provides a new key constraint on a combination of parameters related to the origin and assembly for the most massive BHs: formation efficiency and clustering, growth efficiency, and the relative contribution of BH mergers. Methods. We observed 116 spectroscopically confirmed QSOs around redshift 6 with the simultaneous seven-channel imager Gamma-ray Burst Optical/Near-infrared Detector in order to search for companions. Applying colour-colour cuts identical to those which led to the spectroscopically confirmed QSOs, we performed Le PHARE fits to the 26 best QSO pair candidates, and obtained spectroscopic observations for 11 of them. Results. We do not find any QSO pair with a companion brighter than M₁₄₅₀(AB) < −26 mag within our 0.1–3.3 h⁻¹ cMpc search radius, in contrast to the serendipitous findings in the redshift range 4–5. However, a small fraction of such pairs at this luminosity and redshift is consistent with indications from present-day cosmological-scale galaxy evolution models. In turn, the incidence of L- and T-type brown dwarfs, which occupy a similar colour space to z ∼ 6 QSOs, is higher than expected, by a factor of 5 and 20, respectively.