I picked up a Unique 22 just like the OP's a couple years ago. It was at a gun show, and the price looked pretty cheap. I knew about Unique, as my Dad brought back a Unique 32 from WWII.
When I looked at it, I immediately noticed that the extractor was missing.
When I pointed it out to the seller (a private individual with a table, not a dealer...), he was dumbfounded, and said he had shot a few hundred rounds through it without a jam...
He lowered the price to $80, and I bought it.
I knew it was possible that it would work without an extractor, the same way a Beretta Bobcat does.
Sure enough, a trip to the range found the Unique worked perfectly.
I've put maybe 200rds through it without a jam, and I have found it to be pretty accurate.
What a great, all steel little 22. I highly recommend it.