It is not possible, and it is not a good idea.
It is not possible, because fighter aircraft are very hard to control at low speeds. Even though some aircraft are hover-capable, air-to-air refueling takes a long time and requires incredible precision. Holding the position required to connect to the boom or drogue is hard enough at 270 knots, where the aircraft is easier to control—doing it at low speeds or during a hover would be nearly impossible. Further, maintaining connections for the 5-10 minutes required to transfer fuel would be even worse. Finally, this would force the aircraft down to a hover-able height (for F-35s this is apparently at 10,000 feet or lower), which would cost the plane valuable fuel, time and energy when it had to climb back up to 25-30,000 feet.
It is not a good idea, because tankers already excel at the job and can cover more requirements than just refueling. First of all, tankers can carry an enormous amount of fuel. A KC-10 Extender can carry 365,000 lbs of fuel, that's (exactly) 20 full tanks for an F-35A. Second, they have decent cruise speeds, long range and high station time. Finally, they can carry other cargo than just fuel. These facts combined means that when a squadron deploys to a theater, they can be accompanied by a tanker for the entire flight. This will not only allow the tanker to extend (heh) the practical range of the fighters it accompanies, but it can also carry personnel and equipment required for the squadron that will not fit in the fighters. This is exactly how squadrons were deployed to the Middle East for Desert Shield, for example.
Finally, the dirigible-based tanker would be an even easier target than a tanker aircraft, and would likely be detected and destroyed by hostile forces more or less immediately after hostilities broke out. Due to its low speed, it would be easy for a missile's radar to pick it out and target it specifically, and evasion would be impossible.