The search for extraterrestrial life is one of the most exciting frontiers in present-day astronomy. Recently, the TRAPPIST-1 star was discovered to host seven rocky planets with masses and radii similar to those of the Earth, of which at least three of them may be capable of supporting life. Our paper addresses the possibility that life on one of these planets can spread to others through the transfer of rocky material. We conclude that this process has a high probability of being operational, implying that this planetary system may possess multiple life-bearing planets. Thus, our work has profound theoretical and observational consequences for future studies of the TRAPPIST-1 system and the likelihood of life in our galaxy.