Can Lillia Fae Fawn move from a battlefield where she is the only unit and maintain control of the battlefield with the Sprite she creates from moving?

Yes, Lillia, Fae Fawn can move from a battlefield where she is the only unit and maintain control of it.

According to RiftJudge FAQ #10369, when Lillia moves from a battlefield, her triggered ability places a 3 [M] Sprite token (with [Temporary]) at the location she just left. This sequence of events creates a Closed State on the chain. Per Rule 187.4.c, you cannot lose control of a battlefield while the turn is in a Closed State. Because the Sprite token is created as part of the resolution of the ability that triggered on the move, you will maintain control of the battlefield throughout the process, and the newly played Sprite token will be present there to establish control once the chain is empty and the state returns to "Open."

Sequence Summary:

1. Lillia moves from the battlefield.
2. Her triggered ability ("When I move from a location, play a 3 [M] Sprite token...") is added to the chain.
3. The game enters a Closed State.
4. Because the state is Closed, you do not lose control of the battlefield, even though no units are currently there.
5. The ability resolves, playing the Sprite token at that battlefield.
6. Once the chain is empty and the state returns to "Open," the Sprite token is present at the battlefield, allowing you to maintain control.