  The baryonic form factor of charged pions is studied in detail in the
Nambu--Jona-Lasinio model with constituent quarks, where the spontaneously
broken chiral symmetry is the key dynamical ingredient guaranteeing the
would-be Goldstone boson nature of the pseudoscalar mesons octet. In general,
this form factor arises when the isospin symmetry is broken, which is the case
if the $u$ and $d$ quark masses are split, as in the real world, or if the
electromagnetic effects were taken into account. We obtain estimates for this
basic property of the pion resulting from the quark mass splitting for a range
of model parameters, and importantly, for different pion masses, going up to
the values used in lattice QCD. We find very stable model results, with the
mean square radius of $\pi^+$ in the range $(0.05-0.07~{\rm fm})^2$. From
charge conjugation, the baryonic form factor of $\pi^+$ and $\pi^-$ are equal
and opposite. We also obtain the transverse-coordinate, relativistically
invariant baryonic density of the charged pion. In $\pi^+$, the inner region
carries a negative, and the outside -- a positive baryon number density, both
cancelling to zero, as obviously the pion carries no net baryon charge. We also
carry out an analogous analysis for the kaon, where the effect is much larger
due to the sizable $s$ and $u/d$ quark mass splitting. We discuss the prospects
of lattice QCD measurements of the baryonic form factors of charged pions and
kaons.
