Measurements of proton-induced dark current increases in a Si CID (charge injection device) imager have been made following displacement damage by 12- and 63-MeV protons. Populations of 61,504 pixels optimize statistics and make possible the first detailed study of rare events. To this end, extreme value statistics allow a quantitative treatment and lead to characterization of a rare device-dependent mechanism. Data comparing the response of two similar CID structures suggest that electric-field-enhanced emission is responsible for the largest dark current increases in the CID structure with the higher electric fields. Comparisons between observations and estimates based on new calculations of the recoil spectrum parameters demonstrate that the largest dark current increases can be predicted in the absence of high fields. In this case the inelastic recoil component of the recoil spectrum plays a dominant role in determining the large dark current increases. Implications for other material are discussed.