> the people who provide the dollars to this market (banks, I'm informed)
You are misinformed. The people who provide the dollars to this market are the customers of the exchanges. It seems like you are not familiar with how a currency exchange works.
A currency exchange does not buy or sell the currencies traded at the exchange. All it does is match buyers with sellers, like an auction house. Buyers must deposit all the dollars that they use to buy before buying, and sellers must deposit all the bitcoins they sell before selling. When someone withdraws dollars after selling, the exchange simply transfers to them the dollars that the buyer deposited earlier. It already has the dollars on hand and does not need to borrow them from a bank, ever.
Now, not all places you can sell Bitcoin for dollars are currency exchanges. There are also brokers that do trade directly with their customers, rather than matching customers together to trade. In fact, Coinbase provides this service. However, they can simply obtain dollars by trading at currency exchanges themselves, and they adjust the price they offer to match the exchange price (plus a fee) so they can't just run out of money. Again, there is no need to borrow dollars from a bank. Furthermore, trading volumes at true currency exchanges absolutely dwarf the volume of any Bitcoin broker.
All that said, nothing prevents a crash in the Bitcoin price. If too many people want to sell and not enough want to buy, the price will go down a lot. But this is true in any market, and it has nothing at all to do with bank lending. Not even a little bit.