Does a creature with lifelink gain life when it deals damage to a planeswalker?

Lifelink is a static ability that causes damage dealt by the creature to also cause its controller to gain that much life. Crucially, this applies to any damage the creature deals, not just combat damage to players (CR 702.15a).

Planeswalkers are legal targets for combat damage and certain other damage sources. When a creature with lifelink deals damage to a planeswalker, that damage removes loyalty counters from the planeswalker (CR 306.7) and simultaneously triggers the lifelink life-gain effect (CR 702.15b).

It is worth noting that lifelink's life gain is not a triggered ability — it is a replacement-style static ability that happens as part of the same event as the damage itself (CR 702.15b). This means it cannot be responded to separately; the life is gained at the same time the damage is dealt.

Example: You attack with a 3/3 creature with lifelink. Your opponent blocks with a planeswalker (by redirecting your attack, or in a format that allows attacking planeswalkers directly). Your creature deals 3 damage to the planeswalker, removing 3 loyalty counters — and you simultaneously gain 3 life due to lifelink.

The same principle applies if the creature deals non-combat damage to a planeswalker, such as via a fighting effect or a damage-dealing activated ability. As long as the source with lifelink is dealing the damage, its controller gains life equal to the damage dealt.