Can you cast a spell at instant speed during the cleanup step?

Under normal circumstances, no player receives priority during the cleanup step, which means no one can cast spells or activate abilities. The cleanup step is designed to be automatic: the active player discards down to their maximum hand size, then all damage is removed from permanents and 'until end of turn' effects end. (CR 514.1, 514.2)

However, there is an important exception: if any state-based actions need to be performed, or if any triggered abilities trigger during the cleanup step, the game pauses. Players receive priority in a new cleanup step after those triggers resolve, and during that window you can cast instants or activate abilities. (CR 514.3)

This means the answer is conditional: you can cast a spell at instant speed during cleanup only if a triggered ability or state-based action creates a priority window. Without that, the step simply ends and the game moves to the next turn.

Example: Your opponent controls a Kavu Predator with a triggered ability that triggers whenever you gain life. You have a Gilded Lotus and use a Recoup to flashback a spell that gains you life at end of turn — it triggers in cleanup. Because Kavu Predator's ability now triggers during cleanup, a new priority window opens after it resolves, and you could cast a Counterspell or other instant during that window.

So in practice, most cleanup steps end without any player ever getting priority. If you want to create a window to cast an instant in cleanup, you need to engineer a trigger or state-based action during that step.