If I target a unit at a battlefield with piercing shot and then a unit in base, if my opponent kills the unit at the battlefield with hidden blade, does the unit in base still take damage?

Yes, the unit in the base will still take 2 damage.

According to RiftJudge FAQ #813 and #9000, the "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.

Sequence of Events:

1. You play Piercing Light: You declare your targets (one at a battlefield, one at a base) and put the spell on the chain.
2. Your opponent responds: They play Hidden Blade (presumably from a Hidden state) targeting your unit at the battlefield.
3. Chain resolves (LIFO - Last In, First Out):
   • Hidden Blade resolves first: It kills your unit at the battlefield.
   • Piercing Light resolves next: Even though its first target is now gone (it was killed), the spell proceeds to the second part of its instruction and deals 2 damage to your second target (the unit in the base).

As per Rule 359.3.e.8, "If an instruction has more than one Target and fewer than all of the Targets become Invalid or Unavailable by the time the spell is finished being played, the instruction will execute, with only the Targets available and valid being operated on."