A continuous well string injector features a frame carrying a pair of endless chain conveyors on opposite sides of a pathway. Each chain carries gripper dies for engaging the continuous well string when received in the pathway. A respective skate and cylinder block reside in the closed loop path of each chain. Each cylinder block features a plurality of blind bores having open ends facing toward the respective skate, and a respective piston is slidably disposed within each bore. Hydraulic extension and retraction ports open into the cylinder bores on opposite sides of the piston, whereby back and forth displacement of the pistons forces the skates toward and away from one another to grip and release the continuous well string. A front wall of the frame has an open horizontal gap aligned with the pathway to enable lateral entrance and exit of the continuous string to and from the pathway.