The combinatorial clock auction (CCA) has frequently been used in recent spectrum auctions. It combines a dynamic clock phase and a one‐off supplementary round. The winning allocation and the corresponding prices are determined by the Vickrey–Clarke–Groves rules. These rules should encourage truthful bidding, whereas the clock phase is intended to reveal information. We inquire into the role of the clock when bidders have lexicographic preferences for raising rivals' costs. We show that in an efficient equilibrium, the clock cannot fully reveal bidders' types. In the spirit of the ratchet effect, in the supplementary round competitors extract surplus from strong bidders whose type is revealed. We also show that if there is substantial room for information revelation, that is, if the uncertainty about the final allocation is large, all equilibria of the CCA are inefficient. Qualitative features of our equilibria are in line with evidence concerning bidding behavior in some recent CCAs.