If combat starts and I use Challenge on my unit at base against a unit at the Showdown, it dies. Combat ends. Does my base unit heal?

Yes, your unit at the base heals.

According to RiftJudge FAQ #6111, when a combat showdown concludes, all units currently in play heal all damage on them simultaneously, regardless of whether they participated in the combat or where they are located.

Here is the breakdown of the sequence:

1. Combat Initiation: A combat showdown is ongoing at a battlefield.
2. Challenge: You play Challenge targeting your unit at the base and an opposing unit at the battlefield.
3. Resolution: The targeted units take damage. If your unit at the base takes lethal damage, it is killed.
4. Combat Conclusion: The showdown continues until both players pass, at which point the showdown ends.
5. Combat Cleanup: As part of the combat resolution process, the game performs a "Combat Cleanup" (Rule 461.1). This step includes healing all units in play (Rule 461.1.a.1).

Because a combat showdown was initiated and subsequently concluded, the global healing effect triggered by the "Combat Cleanup" step applies to all units in play, including any surviving units at your base.

Note: If your unit at the base was killed by the Challenge effect, it is no longer "in play" and therefore cannot be healed.