This letter reports rich X-ray jet structures found in the Chandra observation of PKS 1055+201. In addition to an X-ray jet coincident with the radio jet we detect a region of extended X-ray emission surrounding the jet as far from the core as the radio hotspot to the North, and a similar extended X-ray region along the presumed path of the unseen counterjet to the Southern radio lobe. Both X-ray regions show a similar curvature to the west, relative to the quasar. We interpret this as the first example where we separately detect the X-ray emission from a narrow jet and extended, residual jet plasma over the entire length of a powerful FRII jet.