Flip the chair over so the casters are facing up. Pry up the clip with a flat-head screwdriver and slide it out. Remove the washer, then lift off the casters. Lift off the plastic housing over the pneumatic cylinder.
Spray water-displacing lubrication such as WD-40 around the cylinder as close to the seat of the chair as possible. Firmly strike the cylinder with a hammer from all four sides to provide space for the lubricant to work down into the mechanism. Wait five minutes.
Tighten a pipe wrench around the cylinder as close to the seat as possible. Alternate turning the cylinder clockwise and counterclockwise while lifting up and away from the seat. The cylinder is held in with tapered metal, not screw threads. Work it out like a cork from a bottle. Be firm and patient. Don't force the cylinder because you might break the mechanism in the chair. If you can't get enough grip with the cylinder attached to the seat, unscrew the mechanism from the bottom of the seat and use a bench vise or other clamp to hold the mechanism firmly as you twist out the cylinder.
Insert the tapered end of a new cylinder into the mechanism. Fit it snugly with your hands (no tools). Replace the plastic cover and the casters. Reinstall the washer and then slide the clip in place. Sit down on the chair several times to fully seat the cylinder. Press the lever to raise and lower the cylinder three or four times.