Does lifelink apply when a creature with lifelink 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 (CR 702.15a). The rule applies to all damage the creature deals — to players, planeswalkers, other creatures, or anything else that can be dealt damage.

When a creature with lifelink deals damage to a planeswalker, two things happen simultaneously: the planeswalker loses that many loyalty counters (CR 120.3c), and the attacking creature's controller gains that much life (CR 702.15a). There is no restriction limiting lifelink to damage dealt only to players or creatures.

It is worth noting that under the current rules (since the 2019 rules update), damage dealt to planeswalkers is damage dealt to the planeswalker's controller for the purposes of effects that care about damage to players — but lifelink does not need that bridge; it simply triggers on any damage dealt by the creature (CR 120.3).

Example: You attack with a 3/3 creature with lifelink. Your opponent blocks with nothing, and you redirect the damage to their Jace, the Mind Sculptor with 4 loyalty. Jace loses 3 loyalty counters (going to 1), and you gain 3 life from lifelink.