Is there a charging cable for the Apple TV remote?

Yes. It is sold separately.

Which cable you use depends on which Siri Remote you have.

The original and the 2nd generation have a Lightning port.

The 3rd generation has a USB-C port.

Also see: Charge the Siri Remote - Apple Support

For the first two you can use any Lightning to USB or USB-C cable.

For the 3rd generation you can use a USB-C to USB-C cable.

You may already have a cable if you have an iPhone or iPad that uses the same cable.

Not a specific one for it, but most charging cables used with an iPhone or iPad should work.

If its the newer USB-C remote, a cable from an iPhone 15 or recent MacBook (Pro or Air), iPad Pro or iPad Air will work for it.

click here ➜ Charge the Siri Remote - Apple Support