  Recently, four additional Earth-mass planets were discovered orbiting the
nearby ultracool M8 dwarf TRAPPIST-1, making a remarkable total of seven
planets with equilibrium temperatures compatible with the presence of liquid
water on their surface. Temperate terrestrial planets around an M-dwarf orbit
close to their parent star, rendering their atmospheres vulnerable to erosion
by the stellar wind and energetic electromagnetic and particle radiation. Here,
we use state-of-the-art 3D magnetohydrodynamic models to simulate the wind
around TRAPPIST-1 and study the conditions at each planetary orbit. All planets
experience a stellar wind pressure between $10^3$ and $10^5$ times the solar
wind pressure on Earth. All orbits pass through wind pressure changes of an
order of magnitude and most planets spend a large fraction of their orbital
period in the sub-Alfv\'enic regime. For plausible planetary magnetic field
strengths, all magnetospheres are greatly compressed and undergo much more
dynamic change than that of the Earth. The planetary magnetic fields connect
with the stellar radial field over much of the planetary surface, allowing
direct flow of stellar wind particles onto the planetary atmosphere. These
conditions could result in strong atmospheric stripping and evaporation and
should be taken into account for any realistic assessment of the evolution and
habitability of the TRAPPIST-1 planets.
