If I use Relentless Pursuit on a unit at Vilemaw's Lair, can I move that unit to the other battlefield?

Yes, you can move a unit from Vilemaw's Lair to another battlefield using Relentless Pursuit.

The restriction on Vilemaw's Lair specifically states: "Units can't move from here to base." It does not prevent units from moving between battlefields (often referred to as "Ganking" when done via a Standard Move, or simply moving via a spell effect).

Key Details:

• Destination Matters: Vilemaw's Lair only blocks the "Base" as a valid destination for a move. Since you are moving the unit to the "other battlefield," the restriction does not apply.
• Relentless Pursuit Effect: The spell allows you to move a friendly unit. As long as the destination is not your base, the move resolves normally.
• Conquer Trigger: Note that Relentless Pursuit grants the unit an ability: "When I conquer, you may move me to my base." If the unit moves to the other battlefield and conquers it, that specific triggered move will fail because Vilemaw's Lair's restriction (preventing moves to base) would still apply if the unit is still at the Lair (though in this scenario, the unit has already moved to the other battlefield, so it would be able to move to base from that new location).

Related Ruling: According to official FAQ guidance regarding Vilemaw's Lair: *"Vilemaw's Lair only prevents moving from Vilemaw to base. It... does not prevent moving from Vilemaw to another battlefield."