When Lilia Fae Fawn is the only unit at a battlefield and moves from a battlefield to base, do I still have control of the battlefield from the sprite that gets created (so no hidden card gets removed)?

Yes. When Lillia, Fae Fawn moves from a battlefield where she's the only unit, you maintain control — the Sprite token she creates will be there to keep it, and any hidden card stays put.

Here's why: Lillia's triggered ability ("When I move from a location, play a 3 Sprite token there") goes on the chain as soon as she moves. Per Rule 187.4.c, control of a battlefield cannot be lost while the turn is in a Closed State (i.e., while there are items on the chain). So the moment she moves and her trigger becomes pending, you're protected — the Sprite token then resolves and is present at that battlefield to establish control once the chain empties and the state opens.

No hidden card is removed, because you never lost control at any point. (FAQ #11108, #10369)