A measurement of the ratio R-chi c = (chi(c) --> J/psi + gamma)/J/psi in pC, pTi, and pW interactions at 920 GeV/c (root s = 41.6 GeV) in the Feynman-x range -0.35 < x(F)(J/psi) < 0.15 is presented. Both mu(+)mu(-) and e(+)e(-) J/psi decay channels are observed with an overall statistics of about 15 000 chi(c) events, which is by far the largest available sample in pA collisions. The result is R chi(c) = 0.188 +/- 0.013(st-0.022sys)(+0.024) averaged ratio R-12 = R-chi c1/R-chi c2 is measured to be 1.02 +/- 0.40, leading to a cross section ratio sigma(chi(c1))/sigma(chi(c2)) = 0.57 +/- 0.23. The dependence of R-chi c on the Feynman-x of the J/psi, x(F)(J/psi), and its transverse momentum, p(T)(J/psi), 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-chi c and R-12 due to the dependence of acceptance on the polarization states of J/psi and chi(c) is performed. By varying the polarization parameter, lambda(obs), of all produced J/psi's by two sigma around the value measured by HERA-B, and considering the maximum variation due to the possible chi(c1) and chi(c2) polarizations, it is shown that R-chi c could change by a factor between 1.02 and 1.21 and R-12 by a factor between 0.89 and 1.16.