Observing the famous high-mass, eccentric X-ray and gamma-ray binary PSR B1259-63/LS 2883 with Chandra, we detected X-ray emitting clumps moving from the binary with speeds of about 0.1 of the speed of light, possibly with acceleration. The clumps are being ejected at least once per binary period, 3.4 years, presumably around binary periastrons. The power-law spectra of the clumps can be interpreted as synchrotron emission of relativistic electrons. Here we report the results of 8 observations of the clumps in 2011-2017 (two binary cycles) and discuss possible interpretations of this unique phenomenon.