The TRAPPIST-1 system is unique in that it has a chain of seven terrestrial Earth-like planets located close to or in its habitable zone. In this paper, we study the effect of potential cometary impacts on the TRAPPIST-1 planets and how they would affect the primordial atmospheres of these planets. We consider both atmospheric mass loss and volatile delivery with a view to assessing whether any sort of life has a chance to develop. We ran N-body simulations to investigate the orbital evolution of potential impacting comets, to determine which planets are more likely to be impacted and the distributions of impact velocities. We consider three scenarios that could potentially throw comets into the inner region (i.e. within 0.1 au where the seven planets are located) from an (as yet undetected) outer belt similar to the Kuiper belt or an Oort cloud: planet scattering, the Kozai–Lidov mechanism, and Galactic tides. For the different scenarios, we quantify, for each planet, how much atmospheric mass is lost and what mass of volatiles can be delivered over the age of the system depending on the mass scattered out of the outer belt. We find that the resulting high-velocity impacts can easily destroy the primordial atmospheres of all seven planets, even if the mass scattered from the outer belt is as low as that of the Kuiper belt. However, we find that the atmospheres of the outermost planets f, g, and h can also easily be replenished with cometary volatiles (e.g. ~ an Earth ocean mass of water could be delivered). These scenarios would thus imply that the atmospheres of these outermost planets could be more massive than those of the innermost planets, and have volatiles-enriched composition.