Could a drone physically intercept an incoming missile?

Could a drone protect a ship by flying into an incoming missile after its other defenses were exhausted?

In principle, yes. The collision idea is possible, and a force with AI able to seize a crewed fighter should logically field uncrewed aircraft for the same sacrificial task.

The immediate answer is that an uncrewed aircraft could be placed in a missile's path as an emergency physical barrier. A science-fiction novel supplies the example: an AI takes control of a British-operated F/A-18 and sacrifices the crewed aircraft to protect a carrier. The criticism is institutional rather than cinematic. A system trusted to command a crewed combat aircraft should also be capable of directing an expendable uncrewed platform. The example shows how fiction can expose a force-design inconsistency when autonomy, personnel risk, and available hardware do not develop together.