Buckmark - Troubleshooting

Basically (in easiness order... checks first, cleaning second, changes third):

1. Make sure the rear sight base screw is tight (the one at the top, behind the rear sight).
2. Make sure the barrel screw is tight (unlikely to be loose, but easy to check)
3. Check to see if the firing pin has put a dent in the back of the barrel, above the chamber that is so bad that it has pushed some metal down into the chamber area, causing the interference for rounds going in and out of the chamber.
4. Do the plunk test. Hold the gun (or just the barrel) with muzzle down, drop a round in the chamber. Gravity alone should fully chamber the round, if you can push it farther into the chamber with your finger, it is failing the plunk test. Then flip it so the muzzle is up, and the round should fall out. If it does not fall out, it is failing the test.
5. Check to see if the slide moves easily (when the hammer is cocked). It will have some resistance, because you are compressing the recoil spring, but it shouldn't feel like it's binding, grinding, or hitting stuff as it moves back and forth.
6. Verify that the extractor is there, the hook looks unbroken, and it moves left-right as expected. Buckmarks are blowback operated, so the extractor isn't responsible for pulling the round out of the chamber, but it's easy to check, and the round could be getting blown out and pushed back in as the slide cycles.
7. Verify that the ejector is there and looks ok. This is the wire that sticks out the front of the slide at the bottom left when the slide is locked back. It should be sticking out near the top-right corner of the hole it emerges from, and the part that you can see should be straight (the whole part is bent into an S-shape and staked to the frame, but you can't see that until you remove the slide).
8. Clean the chamber (usual suggestion is to use a .25 or .24 diameter rifle brush bent into an L-shape so that you can only stick it in as far as the shell casing goes). You want to make sure there is no carbon build-up, such as rings, in the chamber.
9. Clean the gun in general, and lube where the slide rubs the frame. Partly this is to remove the preservative applied at the factory and replace it with actual lubricant, partly this is to check for anomalies, and partly this is just good practice. Try cleaning what you can without removing the slide first. If FTEs persist, take off the slide and clean more thoroughly.
10. Try different ammo. In particular, if you've been using standard velocity (SV) ammo, try shooting a few hundred high-velocity (HV) rounds. Sometimes the springs are a bit stiff when new, and sometimes the parts just need to wear in a little before they run as smoothly as they should. This is a generic gun issue, not specific to the Buckmark.

You can usually bend it back into shape by hand, roll it on a table to see if it is pretty straight, and put it back in.

RFC Thread 515573 about firing pin shape.
Recommends only hitting inside the rim (the idea being that the edge of the rim is just bent brass, no primer, and crushing that is a useless waste of FP energy)
Also recommends that the tip be angled back slightly... extending further at the bottom (toward the center of the case), and extending less near the rim, so as to match the angle/shape of the part of the casing you are crushing the primer against.
Also recommends Pin tip width: .025" and Pin tip height: .075"
Quoting

"Eley recommends about .015" dent depth for their ammo, most custom makers like to see at least .018" "

I also see recommendations for pin tip protrusion of 0.025" - 0.035"
BenchRest.com thread - has good images of firing pin shapes
RFC Thread 5161400 - has drawings and dimensions for stoning a firing pin RFC Thread 243340 - another, but less useful thread about firing pin shapes