There's a few popular exchanges that provide this already...haven't used an exchange in a while so I'm not sure which ones nor can I recommend any however this wouldn't be hard to find out.
When gold was the currency there were loans. Even fractional reserve loans.
Lender has 1 oz of gold. He issues a certificate to 10 different people in the amount of 1 oz of gold. People use those certificates as if they are money. At some point someone can claim their 1 oz of gold but they count on not many people doing so.
Bitcoin company issues a Bitcoin credit card, they have 1 BTC and offer 1 BTC worth of credit to 10 different people. The person uses his Bitcoin credit card wherever that credit card is accepted. The merchant then gets a "credit" for however many bitcoins were spent on their merchant account (just a value in the database). The merchant can withdraw their bitcoins or use their own Bitcoin credit card to make further purchases.
They rely on not everyone wanting to take their bitcoins out at once. Otherwise they are screwed, that is why a bank run was always such a big deal to banks...in Mary Poppins they gave a glimpse of what would happen if people wanted their money from the bank all at once.
When it comes to credit limits and whether people pay the bitcoins back, they would be subject to the same credit rating as other credit cards. If you're late, your credit score goes down.