Question is really can you pay or receive a fraction of a satoshi?
Not at all, satoshi is the smallest unit of a bitcoin which cannot be further divided into other units and someone can only receive 1 satoshi minimum as a payout from a faucet or something.
Yeah i never heard any name below satoshi and this is i think the smallest the second is ubtc 3rd mbtc. until 1 btc.. but there is other terms for those name .If you want to know more about better to check this wiki for more info about bitcoin or satoshi here
https://en.bitcoin.it/wiki/Units