Can I play Thrill of the Hunt to a battlefield I don't control?

Yes, you can play Thrill of the Hunt to move a unit to a battlefield you do not control.

According to the card's text, when you play Thrill of the Hunt, you banish a friendly unit, and then its owner plays it to "any battlefield." There is no restriction in the card text or the game rules that limits this choice to battlefields you control.

However, keep in mind the following:

• Contested Status: If you play the unit to a battlefield you do not control, that battlefield will become Contested (Rule 187.3.a), which may trigger a showdown.
• Invalid Destinations: In multiplayer games (3+ players), if the battlefield you choose has a combat or showdown already in progress involving players other than yourself, it may be an invalid destination (Rule 457.2). In that case, the unit would be played to your base instead (Rule 457.2.a).