Yes, big cats can get Covid-19. There have been two instances of captive big cats getting Covid-19 concerning both tigers and lions. There was a well-publicised news story during the early parts of the coronavirus pandemic (April 2020) at which time New York City was going through a particularly torrid time with a surge in infections among people. It was reported that a four-year-old female Malayan tiger, Nadia, had tested positive for Covid-19 together with three other tigers, with three African lions showing similar symptoms.
Eight cats were tested using a fecal sample test developed by the zoo’s partners in which animals do not have to be placed under anaesthetic. The testing was done by a veterinary laboratory. As I understand it, there was no danger of the cats infecting people. It is believed that zookeepers infected the cats or cats and then it spread between the cats (link to the story on PoC).

Barcelona Zoo
More recently, in fact today, there is a report by The Daily Mail that four lions have tested positive for Covid-19 at Barcelona Zoo. This is the second known case in which big cats have contracted the coronavirus disease. It concerns three females and one male. The females are named Zala, Nima and Run Run. The male is Kiumbe. Two staff at the zoo tested positive for Covid-19. The outbreak was detected last month. At the moment we are not being told how the lions became infected but it seems likely that it was from zookeepers or a zookeeper.
The Barcelona veterinary services have been in communication with the Bronx Zoo in New York. As is common to both cases the symptoms of the virus in the cats were mild. We are told that the lions were given veterinary care for mild clinical symptoms similar to mild flu in humans. They received anti-inflammatory treatment and monitoring. They did not suffer from the more severe symptoms such as difficulty breathing as can happen in humans suffering from this condition. All the symptoms cleared after fifteen days.
The lions had no contact with the public or other animals at the zoo therefore limiting transmission back to people. We should remember that this is a zoonotic disease so it can be transmitted in both directions i.e. from animals to people and people to animals although you will find that researchers claim that we don’t know for sure whether a cat can (or do) transmit the disease to a person. That’s my current understanding but our knowledge is developing rapidly.
I’m told that the puma i.e. a mountain lion, tested positive for the virus in South Africa (The Daily Mail). Although, strictly speaking, the puma is not technically a big cat as it does not roar but it is physically big although much smaller than the tiger and lion.
SOME MORE ON CORONAVIRUS – NEWS STORIES AND COMMENT AS IT DEVELOPED:

Human treatment for Covid is being used on Cyprus’ cats suffering from Feline Infectious Peritonitis (FIP), a coronavirus

How do I know if my cat has Covid?

Tragedy: 3 snow leopards die of Covid

China: when pets get Covid they are killed. When people get Covid they are treated. Unfair and immoral.

Don’t F***k**g Breathe on My Kitten you unvaxxed selfish POS
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