Can you play an action after an Overzealous Fan has removed your unit?

Yes, you can play an action after an Overzealous Fan has removed your unit from the battlefield.

According to the FAQ regarding Overzealous Fan:

• The showdown continues: Even when a unit is removed from the battlefield by Overzealous Fan's ability, the combat showdown does not end immediately.
• Priority and Actions: Players can continue taking actions (such as playing spells or using abilities) during the same showdown. The showdown only ends when both players pass with focus on an empty chain.

Therefore, if your unit is moved to your base by the Fan, you still have the opportunity to play an action (like Ride the Wind to return the unit to the battlefield, or any other valid action) before the showdown concludes.