This comes up on most Bitcoin threads, and you'd think a google search could answer that, but the results are pretty unclear. Apparently Bitcoinica used to but no longer exists. The lack of a direct answer would suggest no, unless someone here chimes in otherwise.

Shorting isn't possible within the Bitcoin protocol itself, since there's no mechanism for the lender to compel the short-seller to cover and return the borrowed bitcoins. And Bitcoins are usually not kept in custody with a broker that could lend them, since doing so would defeat the purpose of the protocol in the first place, that only you can spend your coins. (Any lending or any other means of paying a bitcoin is spending it as far as the protocol is concerned.)

Such an exchange could exist, if it established its own trustworthiness outside of the Bitcoin protocol. It would have power to spend depositors' bitcoins on their behalf, like a bank for fiat currency does, but unlike a bank there is no higher authority such as a government to appeal to if the bank breaches its trust. The exchange would have to do something really really compelling to entice Bitcoin depositors and earn their trust. This set of problems would appear to be deep enough that a Bitcoin short exchange does not currently exist.