The M dwarf GJ 436 hosts a transiting warm Neptune known to experience atmospheric escape. Previous observations revealed the presence of a giant hydrogen exosphere transiting the star for more than 5 h, and absorbing up to 56% of the flux in the blue wing of the stellar Lyman-α line of neutral hydrogen (H I Lyα). The unexpected size of this comet-like exosphere prevented observing the full transit of its tail. In this Letter, we present new Lyα observations of GJ 436 obtained with the Space Telescope Imaging Spectrograph (STIS) instrument onboard the Hubble Space Telescope. The stability of the Lyα line over six years allowed us to combine these new observations with archival data sets, substantially expanding the coverage of the exospheric transit. Hydrogen atoms in the tail of the exospheric cloud keep occulting the star for 10-25 h after the transit of the planet, remarkably confirming a previous prediction based on 3D numerical simulations with the EVaporating Exoplanet code (EVE). This result strengthens the interpretation that the exosphere of GJ 436b is shaped by both radiative braking and charge exchanges with the stellar wind. We further report flux decreases of 15 ± 2% and 47 ± 10% in the red wing of the Lyα line and in the line of ionised silicon (Si III). Despite some temporal variability possibly linked with stellar activity, these two signals occur during the exospheric transit and could be of planetary origin. Follow-up observations will be required to assess the possibility that the redshifted Lyα and Si III absorption signatures arise from interactions between the exospheric flow and the magnetic field of the star.