If a player playing Piercing Light on a unit on a battlefield and I react with a Hidden Hidden Blade, killing the targeted unit, will the second half of the "then" of Piercing Light go through dealing damage to units in base?

Yes, the second part of Piercing Light will still resolve and deal 2 damage to the unit in your base.

According to RiftJudge FAQ #813, #10942, and #10240, the word "then" in Piercing Light is a timing instruction rather than a condition that requires the first target to remain valid or present at the battlefield for the second part of the effect to resolve. Piercing Light follows "Do As Much As You Can" (DAMAYC) logic.

Sequence of Events:

1. Opponent plays Piercing Light: They declare their targets (a unit at a battlefield and a unit at a base) and place the spell on the chain.
2. You respond: You play Hidden Blade from its hidden state as a [Reaction], targeting the unit at the battlefield.
3. Chain resolves (LIFO - Last In, First Out):
   • Hidden Blade resolves first: It kills the targeted unit at the battlefield.
   • Piercing Light resolves next: Even though its first target is no longer on the board, the spell proceeds to the second part of its instruction and deals 2 damage to the second target you chose (the unit in your base), provided that target is still legal.

This is consistent with Rule 359.3.e.8, which states: "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."