Does a planeswalker that is dealt lethal damage die before its controller can activate a loyalty ability?

State-based actions (SBAs) are checked continuously whenever a player would receive priority, and one of those SBAs states that a planeswalker with 0 or fewer loyalty counters is put into its owner's graveyard (CR 704.5i). Crucially, SBAs are checked before any player receives priority to activate abilities.

Loyalty abilities are activated abilities with a loyalty cost (CR 606.1, 606.3). A player may only activate them when they have priority and the stack is empty during their main phase (CR 606.3). Since SBAs are applied before priority is granted, the planeswalker will already be in the graveyard by the time its controller could attempt to activate a loyalty ability.

It doesn't matter whether the planeswalker took combat damage, was hit by a spell, or had its counters removed by another effect—if it reaches 0 loyalty counters, SBAs immediately remove it the next time SBAs are checked, before anyone gets priority.

Example: Your opponent attacks your Jace, the Mind Sculptor (3 loyalty) with a 5/5. Combat damage is dealt, dropping Jace to 0 loyalty. Before you can respond or activate any ability, SBAs are checked, Jace is put into your graveyard, and then players receive priority. You never get a window to use a loyalty ability to save him.

This is a fundamental difference from creatures: a creature with lethal damage marked on it survives until SBAs are checked (same timing), so the result is the same—no opportunity to act in between taking lethal damage and dying.