Could a small drone electric hot air balloon be made to stay in the air indefinitely?
Use an electric element for heating the air. Powered and tethered from the ground like a kite on a string it would not need a battery.
The other option would be a laser. The baloon could reach migher altitudes without the corresponding increase in wire weight. Of course, it would need some means of holding position. Which it might need anyway - a blimp, for example, will need to be flown actively against a storm when tethered.
Obviously, the baloon needs to be made of some material which can withstand the constant firing of laser.
Microwave power could also be used. IIRC, there was some 1960's idea to power a helicopter that way...yes, here we go: https://en.wikipedia.org/wiki/Beam-powered_propulsion#Non-spacecraft_applications