My son is trying to replace a washer in my Peerless bathroom faucet, because the cold water drips a slow drip continually. It might be a washerless faucet, we just don't know. He took an Allen wrench and removed the cold faucet top. There is a brass thing that sticks up that the handle attaches to. There is a plastic thing that is around the brass stem and we can't get any further. Does anyone have any suggestions?

This is a two-handle faucet. Correct?

At this point, I would try taking a pair of pliers, padding the jaws with a cloth, and turning the brass stem in the "on" direction until is comes out. I hope that this works.

You might not have a faucet that has a washer. Delta bought out Peerless and might have incorporated Delta design in your lavatory faucet.
This is how Delta makes their two handle faucets. The trim might be different but the innards are the same. If, after you get the bonnet packing nut off and find a cartridge down there then you'll have to replace the seats and springs,(#4993). They take the place of washers. Good luck, Tom

That was a bad scan. Let me put up another one.