Recent evidence for microlensing of stars in the Large Magellanic Cloud indicates that a significant part of the dark matter in the Galactic halo may be baryonic and consist of substellar objects. Some mechanism is required to form such a distributed halo of low-mass objects. Here, we investigate the possibility that our Galaxy, and possibly most large galaxies, was once surrounded by a cooling flow. Such a flow is commonly observed around the brightest member of a cluster of galaxies, and is inferred to deposit mostly low-mass objects. tf the dark haloes of the most massive cluster galaxies have a large population of low-mass objects formed in this way, then so may lower mass galaxies, such as our own. We show that a strong cooling-flow phase could have occurred during the formation of our Galaxy. Key words: Galaxy: halo - cooling flows - Magellanic Clouds - dark matter - gravitational lensing.