So far I understand that Bitfinex and Poloniex allow it - I've tried it and can confirm. I read that Liqui.io and possibly Bitmex also allow it - I will check these today. Are there any other exchanges that people are aware of?
Bitfinex and Poloniex are the most liquid and reliable markets. If you want to lend at the spread or slightly better, you can usually find liquidity fairly quickly. I don't use Liqui.io personally, but I do know that they offer margin lending with interest similar to Bitfinex and Poloniex.
Bitmex doesn't have the same sort of P2P lending. You can definitely earn interest income there, but it's a lot more complex. I suggest reading in detail how their perpetual swaps work before "lending" there. Here is a snippet from a blog they published last year:
Trade Flow:
With $10,000, buy 20 Bitcoin at a price of $500.
Transfer a minimum of 0.2 Bitcoin as margin to BitMEX, then sell 10,000 XBTUSD contracts at $500.
Store the remaining Bitcoin securely in cold storage.
Each day you will receive the USD Rate – Bitcoin Rate.
There's a tutorial on how their swaps work in the blog I linked. Good luck.