Well, two different actions- EXTRACTING- pulling a cartridge or case out of the chamber, and EJECTING- flicking the cartridge out of the weapon after it is extracted.
Will it extract an unfired round? How about a fired round?
Start by cleaning- scrub the chamber well with a brush and powder solvent (Hoppes 9, or any one that you like). Rough or dirty chamber may make fired brass stick to the chamber walls, and the extractor slip off it's tiny grip on the case rim.
Next, clean the extractor. Soak in some more Hoppes, scrub under the lip of the extractor- old toothbrush, etc. 22 ammo is notoriously dirty, crud building up under extractor may keep it from getting a bite.
If those two do not fix the problem, 2 other possibles come to mind-
First, the tooth of the extractor has been damaged, bent, or worn. Fixed by replacing extractor. Second, when a rimfire is repeatedly DRY FIRED (snapped with no cartridge in the chamber) it can peen the mouth of the chamber- putting a little burr of metal in the way of pulling out the fired cartridge. Cases may have a bright scratch running the length of the case. If THAT is the problem- it can be fixed. Gunsmith uses a tool that looks like a Stanley nailset to IRON the burr back into place (strangely enough, I actually use a Stanley nailset for this) DO NOT POLISH OFF THE BURR- the metal needs to be put back in place, not removed.
Anyone else with some thoughts? Anyone? Bueller? Anyone?