Images of the site of the Type Ic supernova (SN) 2002ap taken before explosion were analysed previously by Smartt et al. We have uncovered new unpublished, archival pre-explosion images from the Canada–France–Hawaii Telescope (CFHT) that are vastly superior in depth and image quality. In this paper we present a further search for the progenitor star of this unusual Type Ic SN. Aligning high-resolution Hubble Space Telescope observations of the SN itself with the archival CFHT images allowed us to pinpoint the location of the progenitor site on the ground-based observations. We find that a source visible in the B- and R-band pre-explosion images close to the position of the SN is (1) not coincident with the SN position within the uncertainties of our relative astrometry and (2) is still visible ∼4.7-yr post-explosion in late-time observations taken with the William Herschel Telescope. We therefore conclude that it is not the progenitor of SN 2002ap. We derived absolute limiting magnitudes for the progenitor of MB≥−4.2 ± 0.5 and MR≥−5.1 ± 0.5. These are the deepest limits yet placed on a Type Ic SN progenitor. We rule out all massive stars with initial masses greater than 7– 8 M⊙ (the lower mass limit for stars to undergo core collapse) that have not evolved to become Wolf–Rayet stars. This is consistent with the prediction that Type Ic SNe should result from the explosions of Wolf–Rayet stars. Comparing our luminosity limits with stellar models of single stars at appropriate metallicity (Z= 0.008) and with standard mass-loss rates, we find no model that produces a Wolf–Rayet star of low enough mass and luminosity to be classed as a viable progenitor. Models with twice the standard mass-loss rates provide possible single star progenitors but all are initially more massive than 30–40 M⊙. We conclude that any single star progenitor must have experienced at least twice the standard mass-loss rates, been initially more massive than 30–40 M⊙ and exploded as a Wolf–Rayet star of final mass 10–12 M⊙. Alternatively a progenitor star of lower initial mass may have evolved in an interacting binary system. Mazzali et al. propose such a binary scenario for the progenitor of SN 2002ap in which a star of initial mass 15–20 M⊙ is stripped by its binary companion, becoming a 5 M⊙ Wolf–Rayet star prior to explosion. We constrain any possible binary companion to a main-sequence star of ≤20 M⊙, a neutron star or a black hole. By combining the pre-explosion limits with the ejecta mass estimates and constraints from X-ray and radio observations we conclude that any binary interaction most likely occurred as Case B mass transfer, either with or without a subsequent common-envelope evolution phase.