A fluid injector has a solid state actuator and a control piston unit arranged in a recess of an injector body. A transmission pin mechanically couples the actuator to the control piston unit. A first control chamber, coupled hydraulically to a second control chamber, is delimited by the control piston unit and the injector body. A control piston of the control piston unit has an end surface that is coupled to the transmission pin and which delimits the first control chamber. The control piston unit has a control sleeve arranged coaxially with the control piston and an inwardly directed projection that provides a driving coupling between the control piston and the control sleeve after a predefined control stroke of the control piston. The driving coupling results in the control piston influencing a free volume of the first control chamber during an axial movement of the control piston.