Results from exoplanet surveys indicate that small planets (super-Earth size and below) are abundant in our Galaxy. However, little is known about their interiors and atmospheres. There is therefore a need to find small planets transiting bright stars, which would enable a detailed characterization of this population of objects. We present the results of a search for the transit of the Earth-mass exoplanet α Centauri B b with the Hubble Space Telescope (HST). We observed α Centauri B twice in 2013 and 2014 for a total of 40 h. We achieve a precision of 115 ppm per 6-s exposure time in a highly saturated regime, which is found to be consistent across HST orbits. We rule out the transiting nature of α Centauri B b with the orbital parameters published in the literature at 96.6 per cent confidence. We find in our data a single transit-like event that could be associated with another Earth-sized planet in the system, on a longer period orbit. Our programme demonstrates the ability of HST to obtain consistent, high-precision photometry of saturated stars over 26 h of continuous observations.