Had a heavily corroded faucet cartridge that I tried to remove using the advice commonly found on YouTube - clamp it with a pair of locking pliers, then either try to pull straight up, or stick a screwdriver between the arms of the pliers and use the screwdriver as a lever to pull it up. I ended up breaking the top part of the cartridge but it wouldn't budge at all. Then I thought about rigging up a screw jack to pull it out.

Supplies

A short piece of metal bar (I used two pieces of angle aluminum)
Two hex head bolts and nuts (3/16" or 1/4" diameter are good size)
Two small wooden blocks
Electric drill and drill bits
Needle nose locking pliers
Wrench
The original screw that fastens the faucet handle to the cartridge (the faucet screw)

Of course, make sure that the water supply is shut off, and the retainer nut for the faucet cartridge is removed.

Step 1: Making the Screw Jack

Cut two pieces of angle aluminum (just because I happened to have them in my workshop), but any piece of metal with sufficient strength and stiffness will work. I overlapped them to make a U-shaped piece just to make sure it is strong enough and won't bend.

Eyeball the dimension of the faucet and drill 3 holes in the metal bar. Middle hole should be the diameter of the faucet screw. The other two holes should be the diameter of the hex bolt.

Step 2: Setting Up the Screw Jack

Put the faucet screw through the metal bar and screw it into the cartridge, at least 1/4".

Screw the nut onto each bolt, and then insert the bolts into the holes into the metal bar.

Put the piece of wood below each bolt and hand tighten the nut until the metal bar is level and pulling up at the faucet screw.

Step 3: Removing the Cartridge

Alternating between the two sides, use the locking pliers to hold onto the head of the bolt, use a wrench to turn the nut so that it pushes up against the metal bar, making sure that the nut is turned the same amount for each side.

The cartridge will come lose after a few turns of the nut to move it up by 1/8" to 3/16" of an inch.

No muss, no fuss, and no risk of damaging or damaging anything.