Motivated by recent theoretical work suggesting that a substantial fraction of Population (Pop) III stars may have had masses low enough for them to survive to the present day, we consider the role that the accretion of metal-enriched gas may have had in altering their surface composition, thereby disguising them as Pop II stars. We demonstrate that if weak, solar-like winds are launched from low-mass Pop III stars formed in the progenitors of the dark matter halo of the Galaxy, then such stars are likely to avoid significant enrichment via accretion of material from the interstellar medium. We find that at early times accretion is easily prevented if the stars are ejected from the central regions of the haloes in which they form, either by dynamical interactions with more massive Pop III stars or by violent relaxation during halo mergers. While accretion may still take place during passage through sufficiently dense molecular clouds at later times, we find that the probability of such a passage is generally low (≲0.1), assuming that stars have velocities of the order of the maximum circular velocity of their host haloes and accounting for the orbital decay of merging haloes. In turn, due to the higher gas density required for accretion on to stars with higher velocities, we find an even lower probability of accretion (∼10−2) for the subset of Pop III stars formed at z > 10, which are more quickly incorporated into massive haloes than stars formed at lower redshift. While there is no a priori reason to assume that low-mass Pop III stars do not have solar-like winds, without them surface enrichment via accretion is likely to be inevitable. We briefly discuss the implications that our results hold for stellar archaeology.