We report on new VLA radio observations of the distant cluster RX J1347.5-1145, which is the most luminous in X-rays. We aim at investigating the possible presence of diffuse and extended radio emission in this very peculiar system which shows both a massive cooling flow and merging signatures. New low resolution (~18 arcsec) VLA radio observations of this cluster are combined with higher resolution (~2 arcsec) data available in the VLA archive. We discover the presence of a diffuse and extended (~500 kpc) radio source centered on the cluster, unrelated to the radio emission of the central AGN. The properties of the radio source, in particular a) its occurrence at the center of a massive cooling flow cluster, b) its total size comparable to that of the cooling region, c) its agreement with the observational trend between radio luminosity and cooling flow power, indicate that RX J1347.5-1145 hosts a radio mini-halo. We suggest that the radio emission of this mini-halo, which is the most distant object of its class discovered up to now, is due to electron re-acceleration triggered by the central cooling flow. However, we also note that the morphology of the diffuse radio emission shows an elongation coincident with the position of a hot subclump detected in X-rays, thus suggesting that additional energy for the electron re-acceleration might be provided by the submerger event.