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 Lyman-α line. We detect significant flux variations during the transits of both Kepler-444 e 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±1.0 Gyr) 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-444 e 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 outof-transit variations, however, would require the presence of a yet-undetected Kepler-444 g 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 Lyman-α 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 11.2 Gyr. Follow-up of the system at XUV wavelengths will be required to assess this tantalizing possibility.