House cats will play with their food i.e. catch a mouse then let it go and then recatch it over and over again before finally eating it.
Do mountain lions exhibit similar behavior?
Fortunately, I cannot give you an answer based on first-hand experience or, unfortunately, even close personal observation. The two sources, (1) Balanced Ecology, Inc and (2) Defenders of Wildlife suggest that cougars do not play with their prey, at least not the larger of their prey. Cougars eat a variety of animals from rabbits to deer. Defenders of Wildlife says they also eat mice.
The Mountain Lion remains alert to any movement, odor or sound and, when at approximately 50 feet, runs or bounds forward attacking the prey from the back or side. The most common form of attack is to grasp the neck and shoulders with its front paws and claws followed by a deadly bite to the neck. Large prey such as deer will often fall to the ground during such a forceful attack.......
After making a kill, the Mountain Lion will usually drag the kill to a protected area and feed on the shoulder and upper abdomen areas first. If cubs are present, they will feed on soft tissues before continuing to consume other body parts.
Mountain lions are active hunters and may travel long distances in search of food. They hunt alone and attack from behind, breaking the neck of their prey by biting it at the base of the skull. After killing their prey, they will bury it and leave it, coming back to feed on it when hungry.
As to why the domestic cat (which includes feral cats) "play" with their prey, Metaphorical Platypus says:
Cats kill their prey by delivering a neck bite that severs the spinal cord. To do this, they must temporarily release the prey to get at the nape of the neck, but when they do so, they risk the prey escaping or counterattacking. Small animals will defend themselves if they get the chance. Mice, rats, and other rodents can deliver a vicious bite, and birds can peck. A cat has a very short muzzle, and to get close enough to apply the neck bite, she risks injury to her eyes and face from the prey.
A cat will “play” with her prey to tire it out in order to reduce the risk of injury to herself, but she’s not actually playing in the human sense.
Perfect Paws says that part of the training of a kitten is for the mother to bring home injured prey and let the kittens learn how to kill by practicing on the injured prey. Perfect Paws also says that the domestic cat's prey can inflict injuries on the cat.
Catwatching, by Desmond Morris (Crown Publishers, 1987), gives more credence to actual playing. Morris says that the hunting drive is independent of the hunger drive, and that for pampered domestic cats catching a little field mouse or a small bird is a great event and the cat cannot bear to end the chase. Morris says that barn cats only occasionally play with their prey, and gives the teaching reason quoted above from Perfect Paws. Morris also mentions the danger of a rodent bite to the cat, and distinguishes the hit-and- chase "play" prompted by this danger from trap-and-release true play.
From the two sources about the cougar, a need to tire out prey does not seem to apply. The cougar attacks with great momentum and makes a quick kill. Moreover, a deer relative to a cougar is larger than a mouse relative to a cat (and can inflict great damage with its sharp hooves); thus the relative danger to the cougar of playing with a deer is greater than the danger of a cat playing with a mouse. As for the additional explanation of Morris, that a domestic cat wants to prolong a rare and exciting hunting experience, this would not apply to a cougar, for whom hunting is a deadly serious business.
What would be interesting is to look at other species of felis and see if the domestic cat and its closest wild relative (felis sylvestris) are the only ones that play with their prey or whether this behavior extends to other species too, and if so where the dividing line is in terms of size and hunting habits.