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The influence of injected current on magnetization dynamics is investigated in submicrometer-wide strips of out-of-plane magnetized Pt/Co/GdOx. This ultrathin film structure is similar to Pt/Co/AlOx exhibiting highly efficient current-driven domain wall (DW) motion, which has recently been attributed in part to the Rashba effect. In Pt/Co/GdOx, the reduction of the out-of-plane magnetization with increasing injected current is largely due to Joule heating, and the effective transverse Rashba field that tilts the magnetization is estimated to be at most ~0.3 T/(1012 A/m2). Measurements of high-speed DW motion reveal no evidence of a strong Rashba field to suppress DW precession. These findings indicate that the Rashba effect plays a limited or negligible role in current-driven DW dynamics, and alternative torques are required to explain the high efficiency of DW motion in ultrathin heavy-metal/ferromagnet/oxide systems.