Active matter encompasses many-particle systems with self-propelling units, such as flocks of birds or schools of fish. Here, we show how self-propelling domain walls can be realised in a solid-state system when a ferrimagnet is weakly driven out of thermal equilibrium by an oscillating field. This activates the Goldstone mode, inducing a rotation of the antiferromagnetic xy-order in a clockwise or anticlockwise direction, determined by the sign of the ferromagnetic component. Two opposite directions of rotation meet at a domain wall in the ferromagnetic component, resulting in `dynamical frustration', with three main consequences. (i) Domain walls move actively in a direction chosen by spontaneous symmetry breaking. Their speed is proportional to the square root of the driving power across large parameter regimes. (ii) In one dimension (1D), after a quench into the ferrimagnetic phase, this motion and strong hydrodynamic interactions lead to a linear growth of the magnetic correlation length over time, much faster than in equilibrium. (iii) The dynamical frustration makes the system highly resilient to noise. The correlation length of the weakly driven 1D system can be orders of magnitude larger than in the corresponding equilibrium system with the same noise level.