There is a new effect that can limit the lifetime of a weakly non-linear wakefield in a radially bounded plasma. If the drive beam is narrow, some of the plasma electrons fall out of the collective motion and leave the plasma radially, forming a negatively charged halo around it. These electrons repeatedly return to the plasma under the action of the charge separation field, interact with the plasma wave and cause its damping. The lowest-energy halo electrons take the energy from the wave more efficiently, because their trajectories are bent by the plasma wave towards the regions of the strongest acceleration. For correct accounting of the wave damping in simulations, it is necessary to take the simulation window at least twice as wide as the plasma.