> Of course you can. People deposit BTC in your bank, and you then loan that out to others while telling the depositor that they can withdraw at any time.

First, without a change in bank legislation and/or rules, I'm not sure a bank could take bitcoins as a deposit. Second, even if they could, if a bank loaned out more bitcoins than it possessed it would be subject to the possibility of a run on depositor's bitcoins. Finally, unlike fiat currency, the central bank in order to end the run could not produce "temporary" new bitcoins to make depositors "whole". I use quotation marks because sometimes new fiat is not temporary and all depositors in the system are not necessarily made whole.