Can you have a hidden card on a battlefield with no other cards?

Yes — but only temporarily, and only if you controlled that battlefield first.

You can't hide a card at a battlefield with no units to begin with: Hiding requires a Battlefield you control (Rule 811.1.b), and control is established by having units there (Rule 190.4). So the hidden card always starts alongside at least one unit.

Once it's there, if all your units leave or die, what happens depends on the game state:

• During an ongoing Showdown/Combat at that battlefield: you keep control, the hidden card stays, and you can still play it even with no units present (FAQ #6143).
• Outside of combat (Open State): you lose control of the battlefield in the following cleanup if you have no units there (Rule 190.4.c), and the hidden card is removed to your trash.

So a lone hidden card on an empty battlefield can exist, but it's a temporary state — it survives only through combat, then goes away.