ABSTRACT With the Australian Square Kilometre Array Pathfinder (ASKAP) we monitored the black hole candidate X-ray binary MAXI J1535–571 over seven epochs from 2017 September 21 to October 2. Using ASKAP observations, we studied the H i absorption spectrum from gas clouds along the line of sight and thereby constrained the distance to the source. The maximum negative radial velocities measured from the H i absorption spectra for MAXI J1535–571 and an extragalactic source in the same field of view are −69 ± 4 and −89 ± 4 km s−1, respectively. This rules out the far kinematic distance ($9.3^{+0.5}_{-0.6}$ kpc), giving a most likely distance of $4.1^{+0.6}_{-0.5}$ kpc, with a strong upper limit of the tangent point at $6.7^{+0.1}_{-0.2}$ kpc. At our preferred distance, the peak unabsorbed luminosity of MAXI J1535–571 was >78 per cent of the Eddington luminosity, and shows that the soft-to-hard spectral state transition occurred at the very low luminosity of (1.2–3.4) × 10−5 times the Eddington luminosity. Finally, this study highlights the capabilities of new wide-field radio telescopes to probe Galactic transient outbursts, by allowing us to observe both a target source and a background comparison source in a single telescope pointing.