We present Chandra X-ray observations of the powerful radio galaxy 3C 171, which reveal an extended region of X-ray emission spatially associated with the well-known 10-kpc scale optical emission-line region around the radio jets. We argue that the X-ray emission comes from collisionally ionized material, originally cold gas that has been shock heated by the passage of the radio jet, rather than being photoionized by nuclear radiation. This hot plasma is also responsible for the depolarization at low frequencies of the radio emission from the jet and hotspots, which allows us to estimate the magnetic field strength in the external medium. We show that it is likely that both the cold emission-line gas and the hot plasma in which it is embedded are being driven out of the host galaxy of 3C 171 at supersonic speeds. A significant fraction of the total energy budget of the central active galactic nuclei must have been expended in driving this massive outflow. We argue that 3C 171, with its unusual radio morphology and the strong relation between the jet and large amounts of outflowing material, is a member of a class of radio galaxies in which there is strong interaction between the radio jets and cold material in the host galaxy; such objects may have been very much more common in the early Universe.