Skyrmions are topologically protected continuous field configurations that cannot be smoothly transformed to a uniform state. They behave like particles and give origins to the field of skyrmionics that promises racetrack memory and other technological applications. Unraveling the non-equilibrium behavior of such topological solitons is a challenge. We realize skyrmions in a chiral liquid crystal and, using numerical modeling and polarized video microscopy, demonstrate electrically driven squirming motion. We reveal the intricate details of non-equilibrium topology-preserving textural changes driving this behavior. Direction of the skyrmion’s motion is robustly controlled in a plane orthogonal to the applied field and can be reversed by varying frequency. Our findings may spur a paradigm of soliton dynamics in soft matter, with a rich interplay between topology, chirality, and orientational viscoelasticity.