I came to the U.S. from New Zealand on the Visa Waiver program, for a ski vacation. I fully intended to leave before the required 90 days -- except that I collided with another skier, and am now in the hospital in a full body cast. Needless to say, I won't be leaving anytime soon. Is there any way I can get an extension so that I don't have an overstay on my immigration record?
Normally, extensions of stay are not allowed under the Visa Waiver Program (VWP). Medical emergencies (as well as changes in travel schedule beyond your control, such as bad weather or a cancellation by the airline) are exceptions to this, however. U.S. Citizenship and Immigration Services (USCIS) can grant extensions of VWP stays, up to 30 days more, in such circumstances.
The catch is that you are expected to show up in person at a USCIS office, using an INFOPASS appointment. Such extension requests cannot be made by mail. From what you've described, that will not likely be possible for you in the near future (though you'd certainly make a dramatic entrance if you could get someone to wheel you in!)
Your best bet is to hire an immigration attorney, who may be able to arrange matters on your behalf.
Your next best bet is to save all records of your hospital stay so that, next time you request a U.S. visa or other entry, you will be able to provide proof that your overstay was due to circumstances beyond your control. It should not be held against you, assuming you leave the U.S. as soon as is reasonably, medically possible.