I cast Piercing Light, and declare an opponent's target in a BF and a target in my opponent's base. My opponent fight or flights his unit at the BF, does the unit in base still take damage?

Yes, the unit in your opponent's base will still take 2 damage.

According to the RiftJudge FAQ #813 and #1752, the word "then" in Piercing Light is a timing instruction, not a condition that requires the first target to remain valid for the second part of the effect to resolve. Piercing Light follows "Do As Much As You Can" (DAMAYC) logic.

Here is the sequence of events:

1. Play: You play Piercing Light, targeting a unit at a battlefield and a unit at a base.
2. Reaction: Your opponent plays Fight or Flight in response, removing their unit from the battlefield.
3. Resolution: When Piercing Light resolves:
   • The first part of the effect (dealing 2 damage to the unit at the battlefield) fails because the target is no longer present/legal.
   • The spell continues to resolve, and the second part of the effect (dealing 2 damage to the unit at the base) is executed.

As stated in Rule 356.3.e.1: "The spell resolves even if some or all of its targets are illegal." Since the second target is still legal, it takes the damage.