Does damage dealt to a creature with lifelink during the first strike damage step trigger lifelink before regular combat damage?

Lifelink is a static ability that causes life gain to happen simultaneously with the damage being dealt — it is not a triggered ability that goes on the stack. Whenever a source with lifelink deals damage, its controller gains that much life as a replacement-style continuous effect at the exact moment damage is dealt (CR 702.15b).

Combat damage is dealt in two separate steps when any creature has first strike or double strike: the first strike combat damage step (CR 310.1 / 509.1) and the regular combat damage step (CR 511). Creatures with first strike or double strike deal damage in the first step; all other creatures deal damage in the second step.

Because lifelink applies at the moment damage is dealt, if a creature with first strike also has lifelink, the life gain from its first strike damage occurs entirely within the first strike damage step — before the regular combat damage step even begins. This can be relevant when your life total matters (e.g., you are at 1 life and need to survive regular combat damage).

Example: Your 2/2 Knight with first strike and lifelink attacks. Your opponent blocks with a 3/3. In the first strike damage step, your Knight deals 2 damage and you immediately gain 2 life. Then, in the regular combat damage step, the 3/3 deals 3 damage to your Knight (destroying it). The lifelink life gain from first strike is fully resolved before that 3 damage lands.

Note: double strike creatures with lifelink gain life in both damage steps — once during first strike and once during regular combat damage (CR 702.4b, 702.15b).