It's rare, but screws can snap off at the head, particularly with old screws. Reddit user funisher reports that using a cheap drill chuck set on reverse can remove the problem screw easily.
You won't want to use this tip on an expensive drill, as it can damage the chuck, but I wouldn't sweat using it on the $20 drill I bought a few years ago on sale. You can also use a pair of locking pliers to remove the screw if you're not in a hurry.