Does a player get priority after each combat damage assignment or only after the combat damage step ends?

Combat damage is not dealt one assignment at a time. During the combat damage step, all attacking and blocking creatures assign their combat damage simultaneously, and then all of that damage is dealt at once. Only after all damage has been dealt does the game check for state-based actions and then grant priority to the active player. This is governed by CR 509.2 and the general priority rule CR 117.3.

Specifically, CR 509.2 states that each creature assigns its combat damage, and then all of that damage is dealt simultaneously. There is no window between individual damage assignments for players to take actions. The combat damage step, like all steps, ends only after both players have passed priority with an empty stack (CR 500.4).

This means you cannot, for example, cast an instant to save a creature after it has been assigned lethal damage but before another creature's damage is assigned. All assignments happen at once, then damage is dealt, then priority is received.

Concrete example: Your 2/2 and your opponent's 3/3 are in combat. Your opponent also controls a 1/1 blocking your other 2/2. All damage is assigned simultaneously: the 3/3 assigns 3 to your 2/2, and the 1/1 assigns 1 to your other 2/2. All damage is then dealt at the same time. You cannot respond between these assignments — you only get priority after all damage resolves and state-based actions are checked.

The only exception involves first strike and double strike (CR 510): creatures with first strike or double strike deal damage in a separate first combat damage step before the regular combat damage step, giving players a priority window in between the two steps — but still not between individual assignments within a single step.