We demonstrate the possibility to drive an antiferromagnetic domain wall at high velocities by fieldlike Néel spin-orbit torques. Such torques arise from current-induced local fields that alternate their orientation on each sublattice of the antiferromagnet and whose orientation depends primarily on the current direction, giving them their fieldlike character. The domain wall velocities that can be achieved by this mechanism are 2 orders of magnitude greater than the ones in ferromagnets. This arises from the efficiency of the staggered spin-orbit fields to couple to the order parameter and from the exchange-enhanced phenomena in antiferromagnetic texture dynamics, which leads to a low domain wall effective mass and the absence of a Walker breakdown limit. In addition, because of its nature, the staggered spin-orbit field can lift the degeneracy between two 180° rotated states in a collinear antiferromagnet, and it provides a force that can move such walls and control the switching of the states.