Possibly "The Day the Icicle Works Closed" by Frederik Pohl. I read it in The Best of Frederik Pohl, and in the English the flying fish are called skyfish:
The skyfish were about the only tourist attraction Altair Nine had left to offer. From all over the Galaxy sportsmen came to shoot them, with their great porous flesh filled with bubbles of hydrogen, real biological Zeppelins that did not fly in the air but swam it. Before human colonists arrived, they had been Altair Nine's highest form of life. They were so easy to destroy with gunfire that they had almost been exterminated in the inhabited sections; only in the high, cold hills had a few survived. And now...
This also has rented bodies:
Physical transport was expensive and eternally slow. But there was a faster way for a man to travel from planet to planet-practically instantaneous, from one end of the Galaxy to the other. The pattern of the mind is electronic in nature. It can be taped, and it can be broadcast on an electromagnetic frequency. What was more, like any electromagnetic signal, it could be used to modulate an ultrawave carrier. The result: Instantaneous transmission of personality, anywhere in the civilized Galaxy.
The only problem was that there had to be a receiver.
The naked ghost of a man, stripped of flesh and juices, was no more than the countless radio and TV waves that passed through everyone all the time. The transmitted personality had to be given form. There were mechanical receivers, of course-computer like affairs with mercury memory cells where a man's intelligence could be received, and could be made to activate robot bodies. But that wasn't fun. The tourist trade was built on fun. Live bodies were needed to satisfy the customers. No one wanted to spend the price of a fishing broadcast to Altair Nine in order to find himself pursuing the quarry in some clanking tractor with photocell eyes and solenoid muscles. A body was wanted, even a rather attractive body; a body which would be firm where the tourist's own, perhaps, was flabby, healthy where the tourist's own had wheezed. Having such a body, there were other sports to enjoy than fishing.
Oh, the laws were strict about misuse of rented bodies.
But the tourist trade was the only flourishing industry left on Altair Nine. The laws remained strict, but they remained unenforced