We present observations of the eccentric gamma-ray binary B1259-63/LS2883 with the Chandra X-ray Observatory. The images reveal a variable, extended about 4, or about 1000 times the binary orbit size) structure, which appears to be moving away from the binary with the velocity of 0.05 of the speed of light. The observed emission is interpreted as synchrotron radiation from relativistic particles supplied by the pulsar. However, the fast motion through the circumbinary medium would require the emitting cloud to be loaded with a large amount of baryonic matter. Alternatively, the extended emission can be interpreted as a variable extrabinary shock in the pulsar wind outflow launched near binary apastron. The resolved variable X-ray nebula provides an opportunity to probe pulsar winds and their interaction with stellar winds in a previously inaccessible way.