If I lose control of a battlefield by having no more units at it, can I play in response to a Deathknell trigger a unit with Ambush to maintain control of the battlefield?

Partly — you do keep control through the Deathknell trigger, but you can't do it with an Ambush unit.

You never actually lose the battlefield at that moment: control only drops in a cleanup when the turn is in an Open state, and while the Deathknell ability sits on the chain the game isn't in an Open state, so the battlefield stays yours through the trigger (Rule 190.4.c / 323.5). That's why Deathknell effects that play a unit from your trash ("play it here") can legally put it back down at that same battlefield.

Ambush doesn't help you though, because it only grants permission to be played "to a battlefield where you control units." With zero units there it's not a valid location at finalization (Rule 822.3), and without that permission it also gets no Reaction timing — so you can't play it in response at all. Once the chain empties and the cleanup runs with no units of yours there, the battlefield goes uncontrolled as normal.