  Kepler-444 provides a unique opportunity to probe the atmospheric composition
and evolution of a compact system of exoplanets smaller than the Earth. Five
planets transit this bright K star at close orbital distances, but they are too
small for their putative lower atmosphere to be probed at optical/infrared
wavelengths. We used the Space Telescope Imaging Spectrograph instrument
onboard the Hubble Space Telescope to search for the signature of the planet's
upper atmospheres at six independent epochs in the Ly-$\alpha$ line. We detect
significant flux variations during the transits of both Kepler-444e and f
(~20%), and also at a time when none of the known planets was transiting
(~40%). Variability in the transition region and corona of the host star might
be the source of these variations. Yet, their amplitude over short time scales
(~2-3 hours) is surprisingly strong for this old (11.2+-1.0Gyr) and apparently
quiet main-sequence star. Alternatively, we show that the in-transits
variations could be explained by absorption from neutral hydrogen exospheres
trailing the two outer planets (Kepler-444e and f). They would have to contain
substantial amounts of water to replenish such hydrogen exospheres, which would
reveal them as the first confirmed ocean-planets. The out-of-transit
variations, however, would require the presence of a yet-undetected Kepler-444g
at larger orbital distance, casting doubt on the planetary origin scenario.
Using HARPS-N observations in the sodium doublet, we derived the properties of
two Interstellar Medium clouds along the line-of-sight toward Kepler-444. This
allowed us to reconstruct the stellar Ly-$\alpha$ line profile and to estimate
the XUV irradiation from the star, which would still allow for a moderate mass
loss from the outer planets after 11.2Gyr. Follow-up of the system at XUV
wavelengths will be required to assess this tantalizing possibility.
