Are there stray cats in the Vatican City?

Yes, there are stray cats in the Vatican. And they are well cared for! We know this because Fr. Robert Ballecer, SJ (a Jesuit) feeds them and he posts tweets about them. He is an American priest who works as the Assistant for New Evangelization for the Curia of the Society of Jesus in Rome. He is a techie guy with experience in commuting.

Stray cat at the Vatican. Image: Fr. Robert Ballecer, SJ. Click on it for a larger version. Cat is on the balcony!!

He's well known and has a Wikipedia page! I'll let Wikipedia tell you more about him. He is high profile priest who thankfully likes cats and is compassionate towards them. He is active on Twitter which is the source of this photo and video.

Here is the video of him feeding some Vatican stray cats. Rome has hundreds of thousands of stray cats. They are a feature of the city. The Mediterranean countries in general are known for their stray cats because the climate is conducive to survival on the street. For example, there are more stray cats on the island of Cyrus than people. Israel also has a stray cat problem. Same reason:

The problem is a lack of comprehensive TNR programs to sterilise them. 

Vatican City is a city-state surrounded by Rome, Italy, and is the headquarters of the Roman Catholic Church. It covers 0.17 mi².