If I play Piercing Light repeated, target a single unit at a battlefield and one at base, and I kill the one at the battlefield with the first instance, will the second repeated instance not deal any damage to the unit at base, or will it still deal the damage?

The second instance of damage will still be dealt to the unit at the base.

According to FAQ 1752 and FAQ 6490, Piercing Light with Repeat is treated as a single, all-in-one spell resolution. The spell performs its instructions twice during that single resolution, and no cleanup (which would remove units from the board) occurs between these two executions.

Here is the sequence of events:

1. Playing the Spell: You play Piercing Light, pay the Repeat cost, and declare all targets for both executions of the spell.
2. Resolution: The spell begins to resolve.
   • First Execution: The first instance of damage is dealt to the unit at the battlefield and the unit at the base.
   • Second Execution: The second instance of damage is dealt to the chosen targets.
3. Cleanup: Only after the entire spell has finished resolving (both executions) does the game perform cleanup. Any units that have taken lethal damage are then sent to the trash.

Because the unit at the battlefield is not removed from the board until after the entire spell has finished resolving, the second execution of the spell proceeds normally. Even if the first execution dealt lethal damage to the unit at the battlefield, that unit remains on the board until the spell finishes, and the second execution will still successfully deal its damage to the unit at the base.

This is further supported by FAQ 813 and FAQ 6726, which clarify that the "then" in the spell text is a timing instruction, not a condition requiring the first target to remain valid for the second part of the effect to resolve.