Exoplanetary systems are found not only among single stars, but also among binaries of widely varying parameters. Binaries with separations of 100–1000 au are prevalent in the solar neighbourhood; at these separations, planet formation around a binary member may largely proceed as if around a single star. During the early dynamical evolution of a planetary system, planet–planet scattering can eject planets from a star’s grasp. In a binary, the motion of a planet ejected from one star has effectively entered a restricted three-body system consisting of itself and the two stars, and the equations of motion of the three-body problem will apply as long as the ejected planet remains far from the remaining planets. Depending on its energy, escape from the binary as a whole may be impossible or delayed until the three-body approximation breaks down, and further close interactions with its planetary siblings boost its energy when it passes close to its parent star. Until then, this planet may be able to transition from the space around one star to the other, and chaotically ‘bounce’ back and forth. In this paper, we directly simulate scattering planetary systems that are around one member of a circular binary, and quantify the frequency of bouncing in scattered planets. We find that a great majority (70–85 per cent) of ejected planets will pass at least once through the space of it’s host’s binary companion, and depending on the binary parameters about 35–75 per cent will begin bouncing. The time spent bouncing is roughly lognormally distributed with a peak at about 104 yr, with only a small percentage bouncing for more than 1 Myr. This process may perturb and possibly incite instability among existing planets around the companion star. In rare cases, the presence of multiple planets orbiting both stars may cause post-bouncing capture or planetary swapping.