We investigated by means of Multi Unit Spectroscopic Explorer/Very Large Telescope (MUSE/VLT) observations the true size of the coronal line region (CLR) in a local sample of nine active galactic nuclei (AGNs) known for displaying prominent coronal emission. Our analysis show that the CLR is extended from several hundred parsecs to a few kiloparsecs in the lines of [Fe VII] (IP = 99 eV) and [Fe X] (IP = 235 eV). In all cases, the coronal emission is closely aligned along the radio-jet axis and constrained to the limits of the [O III] ionization cone. Besides the nuclear emission, secondary emission peaks in [Fe VII] and [Fe X] are found along the extended emission, with a shallow decrease of the line intensity with increasing distance from the AGN. Both facts suggest the action of an additional excitation mechanism besides nuclear photoionization for the origin of the coronal gas. This is further supported by the fact that in some sources the extended coronal emission accounts for more than 50 per cent of the total emission and by the high degree of gas excitation in the off-nuclear region. A positive trend between the coronal line luminosity and the jet power points to shocks induced by the jet passage as the key mechanism to produce and excite this gas. We provide the first estimate of the [Fe X] coronal gas size, being in the kpc range. Our results stress the importance of the CLR as a key ingredient that should be fully considered in models trying to explain the physics of the narrow-line region in AGN.