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 \sim 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 ^ { -1 } ’ 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 } \times 10 ^ { 22 } H cm ^ { -2 } . The observed 2–10 keV flux of 4 \times 10 ^ { -13 } erg s ^ { -1 } cm ^ { -2 } converts to an unabsorbed X-ray luminosity of 1 \times 10 ^ { 34 } erg s ^ { -1 } 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 .