The big cat is believed to be responsible for killing 11 alpacas owned by a Malibu rancher.
The llama-like animals were discovered slaughtered in the rugged Santa Monica mountain range over the weekend.
But the owner says she does not intend to kill P-45, and is surprised by the anger that the permit has drawn.
The California Department of Fish and Wildlife has issued a so-called "depredation order", which under state law grants a person the right to kill a predator that is doing damage to property, including livestock.
The hunting permit was issued on Monday, and owner Victoria Vaughn-Perling has 10 days to kill the animal before the permit expires.
Ms Vaughn-Perling says that she would rather that the animal be relocated, and that she obtained the permit to draw attention to the issue of mountain lion predation on domestic livestock in the region.
But the move sparked angry responses on social media.
"Sorry the mountain lion is eating your pets but a lions gonna do what a lions gonna do. Save #p45", one Twitter user wrote, urging others to sign a petition to protect the cat.
Officials say that the rancher had tried to take measures to protect her animals - a fence and motion-detecting lights were installed after an earlier attack - but that mountain lions can jump 15ft (4.5m) in the air and only a structure with a roof can provide sufficient protection.
P-45 is a male lion, living among a small number of big cats whose territory runs from near downtown Los Angeles to a boundary about 40m (64km) west, towards the Malibu coastline.
The four-year-old mountain lion is believed to be among 10-15 lions living in the Santa Monica mountains between the 101 Freeway and the Pacific Ocean.
A decision by officials in the US state of California to grant a permit for a well-known mountain lion called P-45 to be hunted has caused outrage.