We have measured and theoretically analyzed a photodissociation spectrum of the CH+ molecular ion in which most observed energy levels lie within the fine-structure splitting of the C+ fragment and predissociate, and where the observed irregular line shapes and dipole-forbidden transitions indicate that nonadiabatic interactions lead to multichannel dynamics. The molecules were prepared in low rotational levels J"=0-9 of the vibrational ground state X (1)Sigma(+) (v"=0) by storing a CH+ beam at 7.1 MeV in the heavy-ion storage ring TSR for up to 30 s, which was sufficient for the ions to rovibrationally thermalize to room temperature by spontaneous infrared emission. The internally cold molecules were irradiated with a dye laser at photon energies between 31 600-33 400 cm(-1), and the resulting C+ fragments were counted with a particle detector. The photodissociation cross section displays the numerous Feshbach resonances between the two C+ fine-structure states predicted by theory for low rotation. The data are analyzed in two steps. First, from the overall structure of the spectrum, by identifying branches, and by a Le Roy-Bernstein analysis of level spacings we determine the dissociation energy D-0=(32 946.7+/-1.1) cm(-1) (with respect to the lower fine-structure limit) and assign the strongest features to the vibrational levels v'=11-14 of the dipole-allowed A (1)Pi state. The majority of the 66 observed resonances cannot be assigned in this way. Therefore, in a second step, the complete spectrum is simulated with a close-coupling model, starting from recent ab initio Born-Oppenheimer potentials. For the long-range induction, dispersion and exchange energies, we propose an analytical expression and derive the C-6 coefficients. After a systematic variation of just the vibrational defects of the four Born-Oppenheimer potentials involved, the close-coupling model yields a quantitative fit to the measured cross section in all detail, and is used to assign most of the remaining features to the dipole-forbidden a (3)Pi state (v'=17-20), and some to the weakly bound c (3)Sigma(+) state (v'=0-2). The model potentials, which reproduce the spectrum and compactly represent the spectroscopic data, should help to predict more accurately C++H scattering in the interstellar medium.