An intriguing and elusive feature of the timelike hadron form factor is the possible presence of an imaginary part associated to rescattering processes. We find evidence of that in the recent and precise data on the proton timelike form factor measured by the BABAR Collaboration. By plotting these data as a function of the 3-momentum of the relative motion of the final proton and antiproton, a systematic sinusoidal modulation is highlighted in the near-threshold region. Our analysis attributes this pattern to rescattering processes at a relative distance of 0.7-1.5 fm between the centers of the forming hadrons. This distance implies a large fraction of inelastic processes in pp interactions, and a large imaginary part in the related e(+)e(-)→pp reaction because of unitarity.