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We present RadioAstron space-based very long baseline interferometry (VLBI) observations of the nearby radio galaxy 3C 84 (NGC 1275) at the centre of the Perseus cluster. The observations were carried out during a perigee passage of the Spektr-R spacecraft on September 21–22, 2013 and involved a global array of 24 ground radio telescopes observing at 5 GHz and 22 GHz, together with the Space Radio Telescope (SRT). Furthermore, the Very Long Baseline Array (VLBA) and the phased Very Large Array (VLA) observed the source quasi-simultaneously at 15 GHz and 43 GHz. Fringes between the ground array and the SRT were detected on baseline lengths up to 8.1 times the Earth’s diameter, providing unprecedented resolution for 3C 84 at these wavelengths. We note that the corresponding fringe spacing is 125 μas at 5 GHz and 27 μas at 22 GHz. Our space-VLBI images reveal a previously unseen sub-structure inside the compact ∼1 pc long jet that was ejected about ten years earlier. In the 5 GHz image, we detected, for the first time, low-intensity emission from a cocoon-like structure around the restarted jet. Our results suggest that the increased power of the young jet is inflating a bubble of hot plasma as it carves its way through the ambient medium of the central region of the galaxy. Here, we estimate the minimum energy stored in the mini-cocoon, along with its pressure, volume, expansion speed, and the ratio of heavy particles to relativistic electrons, as well as the density of the ambient medium. About half of the energy delivered by the jet is dumped into the mini-cocoon and the quasi-spherical shape of the bubble suggests that this energy may be transferred to a significantly larger volume of the interstellar medium than what would be accomplished by the well-collimated jet on its own. The pressure of the hot mini-cocoon also provides a natural explanation for the almost cylindrical jet profile seen in the 22 GHz RadioAstron image.