The Boss Katana-Air desktop amplifier and the Yamaha TRH II wireless (10 and 30 Watt models) both have Bluetooth, battery capability, and built-in wireless receivers for guitar input. The Katana-Air includes the transmitter for your guitar while you must purchase a Line 6 G10 transmitter separately for the Yamaha desktop amp.
I returned a Spark amp which does not have a built-in wireless guitar input nor can it run on batteries. Those weren’t the reasons I returned the amp but both features would have been nice. There are alternatives to both built-in Bluetooth and wireless guitar input. For now, I am back to those.
I wrote about Bluetooth receivers previously. If you have an amplifier you love and it has an auxiliary input you can add an external Bluetooth receiver so you can stream backing tracks or tunes to your amp. I have several older Bluetooth receivers I can use but if I were to purchase a receiver now I would buy a USB rechargeable model for convenience. Fewer cords and cables are my goal.
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The easiest wireless solution for guitar input is to purchase a transmitter/receiver pair starting at about $40 and going up to hundreds of dollars for professional gear. The difference in price between home use and pro gear is due to latency, channels, operating distance, and battery life. The pro gear no doubt has better tone too. But the $40 set I purchased works in my home and sounds all right. I don’t have a guitar cord tethered to the amp which is what I wanted.