Yup,
Then you have to drop into HNL, after dumping tons of fuel, because if you don't the person (in the case that happened to me) will be dead by the time you're abeam Fiji.
It happens more than the general public obviously knows, and in many cases by the time the you hit ground the patient is a corpse. Not a lot of fun, the cabin crew is freaked, the flight crew is forced to make decisions above their pay grade, and it's simply not worth it.
As for doctors, you get this a lot out of the S Pacific Islands and trust me, getting a doctors clearance is just not that difficult.