Your ffl has to keep the 4473 (the form w/your social security info, “have you ever worked as a hired killer?”, etc.) forever. When they go out of business, the paperwork is forwarded to the BATFE.
Actually the FFL must keep the 4473 for twenty years, after that they can destroy them. Also, entering your SS# on the 4473 is an option, but not required. (I never do.)