The presence of an atmosphere over sufficiently long timescales is widely perceived as one of the most prominent criteria associated with planetary surface habitability. We address the crucial question as to whether the seven Earth-sized planets transiting the recently discovered ultracool dwarf star TRAPPIST-1 are capable of retaining their atmospheres. To this effect, we carry out numerical simulations to characterize the stellar wind of TRAPPIST-1 and the atmospheric ion escape rates for all the seven planets. We also estimate the escape rates analytically and demonstrate that they are in good agreement with the numerical results. We conclude that the outer planets of the TRAPPIST-1 system are capable of retaining their atmospheres over billion-year timescales. The consequences arising from our results are also explored in the context of abiogenesis, biodiversity, and searches for future exoplanets. In light of the many unknowns and assumptions involved, we recommend that these conclusions must be interpreted with due caution.