We investigate degradation effects in cryogenic Ge detectors subjected to penetrating gamma radiation and to alpha and electron surface bombardment. Based on an analysis of the collection efficiencies and on time-resolved measurements of the ionization signals, we show that the space-charge following gamma irradiation is strongly inhomogeneous, with a maximum density underneath the collection electrodes. Degradation effects induced by alpha and by electron bombardment strongly suggest a dependence of dead layer phenomena on carrier trapping at defect states in the vicinity of the electrodes. Mechanisms of space-charge formation are discussed, including the role of impact ionization of impurities as compared to the more usually considered effects of trapping.