A measurement of the ratio R χ c = ( χ c → J / ψ + γ ) / J / ψ in p C , p Ti , and p W interactions at 920     GeV / c ( √ s = 41.6     GeV ) in the Feynman-x range − 0.35 < x J / ψ F < 0.15 is presented. Both μ + μ − and e + e − J / ψ decay channels are observed with an overall statistics of about 15 000 χ c events, which is by far the largest available sample in pA collisions. The result is R χ c = 0.188 ± 0.013 st + 0.024 − 0.022 sys averaged over the different materials, when no J / ψ and χ c polarizations are considered. The χ c 1 to χ c 2 production ratio R 12 = R χ c 1 / R χ c 2 is measured to be 1.02 ± 0.40 , leading to a cross section ratio σ ( χ c 1 ) σ ( χ c 2 ) = 0.57 ± 0.23 . The dependence of R χ c on the Feynman-x of the J / ψ , x J / ψ F , and its transverse momentum, p J / ψ T , is studied, as well as its dependence on the atomic number, A, of the target. For the first time, an extensive study of possible biases on R χ c and R 12 due to the dependence of acceptance on the polarization states of J / ψ and χ c is performed. By varying the polarization parameter, λ obs , of all produced J / ψ 's by two sigma around the value measured by HERA-B, and considering the maximum variation due to the possible χ c 1 and χ c 2 polarizations, it is shown that R χ c could change by a factor between 1.02 and 1.21 and R 12 by a factor between 0.89 and 1.16.