  Motivated by the planned measurements of the time-like electromagnetic proton
form factors an exploratory study of the time-like electromagnetic pion form
factor is presented in a formalism which describes mesons as
Poincar\'e-invariant bound states. In the respective quark interaction kernel,
beyond the gluon-intermediated interactions for valence-type quarks,
non-valence effects are included by allowing the pions to couple back in a
self-consistent manner to the quarks. Consequently, the opening of the dominant
$\rho$ decay channel, $\rho \to \pi\pi$, and the presence of a multi-particle
branch cut, setting in when the two-pion threshold is crossed, are included
consistently, first in the correspondingly calculated quark-photon vertex, and
then consequently in the time-like electromagnetic pion form factor. The
obtained results are in agreement with the available experimental data and
provide further evidence for the efficacy of a vector-meson dominance model.
Effects of going beyond the here employed isospin-symmetric limit are
discussed. Last but not least, an outlook is provided on how to include in
Poincar\'e-covariant Faddeev approaches the effects of intermediate resonances
in the calculation of baryon properties.
