  We discuss exclusive elastic double diffractive axial-vector \chi_c(1^{+})
meson production in proton-antiproton collisions at the Tevatron. The amplitude
for the process is derived within the k_t-factorisation approach with
unintegrated gluon distribution functions (UGDFs). We show that the famous
Landau-Yang theorem is not applicable in the case of off-shell gluons.
Differential cross sections for different UGDFs are calculated. We compare
exclusive production of \chi_c(1^+) and \chi_c(0^+). The contribution of
\chi_c(1^+) to the J/\Psi + \gamma channel is smaller than that of the
\chi_c(0^+) decay, but not negligible and can be measured. The numerical value
of the ratio of the both contributions is almost independent of UGDFs modeling.
