it is literally impossible to do fractional reserve lending with Bitcoin. each coin has chain of custody back to the inception by mining reward (blockchain) so you can't send BTC you do not own. an altcoin could be built for fractional reserve lending where the coins are created by some form of consensus by a group of authorities. you could trade BTC and this theoretical altcoin but really we already have digital fiat currency for this purpose.
Fractional reserve lending of bitcoin would happen in the same manner as fractional reserve lending of gold coins.
E.g. a bank attracts deposits of bitcoin (i.e. the bank gets the bitcoin, the customer gets an IOU/note in legal records saying that the bank owes them a bitcoin), lends these bitcoins out, some of the lent bitcoin get deposited in another bank and re-lent, etc. In this manner the on-chain bitcoins travel around faster, and the total money supply (bitcoin + all bitcoin-denominated IOU's) can easily grow.
I would argue that it's not really backed by Bitcoin since it isn't on the block chain. Who is backing the lending by sending real Bitcoins? The bank would never do that they would get scammed constantly.
