  The TRAPPIST-1 system offers the opportunity to characterize terrestrial,
potentially habitable planets orbiting a nearby ultracool dwarf star. We
performed a four-orbit reconnaissance with the Space Telescope Imaging
Spectrograph onboard the Hubble Space Telescope to study the stellar emission
at Lyman-$\alpha$, to assess the presence of hydrogen exospheres around the two
inner planets, and to determine their UV irradiation. We detect the
Lyman-$\alpha$ line of TRAPPIST-1, making it the coldest exoplanet host star
for which this line has been measured. We reconstruct the intrinsic line
profile, showing that it lacks broad wings and is much fainter than expected
from the stellar X-ray emission. TRAPPIST-1 has a similar X-ray emission as
Proxima Cen but a much lower Ly-$\alpha$ emission. This suggests that
TRAPPIST-1 chromosphere is only moderately active compared to its transition
region and corona. We estimated the atmospheric mass loss rates for all
planets, and found that despite a moderate extreme UV emission the total XUV
irradiation could be strong enough to strip the atmospheres of the inner
planets in a few billions years. We detect marginal flux decreases at the times
of TRAPPIST-1b and c transits, which might originate from stellar activity, but
could also hint at the presence of extended hydrogen exospheres. Understanding
the origin of these Lyman-$\alpha$ variations will be crucial in assessing the
atmospheric stability and potential habitability of the TRAPPIST-1 planets.
