We report the discovery by XMM-Newton and Chandra of a hard extended X-ray source (XMM J174540-2904.5) associated with a compact non-thermal radio filament (the Sgr A-E `wisp=1LC 359.888-0.086= G359.88-0.07), which is located within ~4 arcmin of the Galactic Centre. The source position is also coincident with the peak of the molecular cloud, M -0.13-0.08 (the `20 km/s cloud). The X-ray spectrum is non-thermal with an energy index of 1.0 (+1.1 -0.9) and column density of 38 (+7 -11) x 10^22 H/cm2. The observed 2--10 keV flux of 4 x 10^-13 erg/s/cm2 converts to an unabsorbed X-ray luminosity of 1 x 10^34 erg/s assuming a distance of 8.0 kpc. The high column density strongly suggests that this source is located in or behind the Galactic Centre Region. Taking account of the broad-band spectrum, as well as the source morphology and the positional coincidence with a molecular cloud, we concluded that both the radio and X-ray emission are the result of synchrotron radiation. This is the first time a filamentary structure in the Galactic Centre Region. has been shown, unequivocally, to have a non-thermal X-ray spectrum.