Author Fallon, 59, also tweeted the news and photo tribute to her pussy pal.
She wrote on social media: "I have awful news. My smart, funny, feisty, beautiful sidekick of the last 16 & a half years @myleftfang has passed away after a sudden down turn in her health at the weekend.
"My heart is broken in two. I have no idea what I'm going to do without her by my side. #Ollie."
Ollie had been in ill health in April 2019 and had to have her stomach fur shaved so she could have an IV drip inserted which led to recovery.
Unfortunately, the pet fell ill again last weekend and took a turn for the worse.
Gervais - who has no children - has often admitted that he prefers animals to humans and regularly highlights animal rights abuses on his Twitter account.
Speaking about his love of animals, he said: "I prefer animals to people - the world is a much nicer place with their little faces.
"I can't abide to cruelty towards animals in any form. It's completely wrong. We need to be kind to them because they make us better people."