Sure you can! Just not on-chain. Just like a bank can lend out a million dollars without having a single physical dollar in the entire building. As long as the dollars (or Bitcoins) only exist on their own accounting system, they can pull it off.
If you try to withdraw a million dollars, the bank won't let you. The same will happen to Bitcoin if they get their way.
Why would you want to get a loan in paper bitcoin, though? Why not take a loan in paper dollars, and use them to buy bitcoin? This way, you don't have counterparty risk (bank going bankrupt, or practicing fractional reserve with no reserves), while you do borrow
the bitcoin
essentially. You're also increasing the supply of dollars, while decreasing the market supply of bitcoin, driving the price up.