No connection to Joan of Arc. But there is a Benedict connection. Bennie 3

No one seems to be entirely sure when the story of Pope Joan/John started, but the first popular form of the story is that after Pope Leo 4 dies in 855:

John Anglicus, born at Mainz, was pope for two years, seven months and four days, and died in Rome, after which there was a vacancy in the papacy of one month. It is claimed that this John was a woman, who as a girl had been led to Athens dressed in the clothes of a man by a certain lover of hers. There she became proficient in a diversity of branches of knowledge, until she had no equal, and afterwards in Rome, she taught the liberal arts and had great masters among her students and audience. A high opinion of her life and learning arose in the city, and she was the choice of all for pope. While pope, however, she became pregnant by her companion. Through ignorance of the exact time when the birth was expected, she was delivered of a child while in procession from St Peter's to the Lateran, in a narrow lane between the Colisseum and St Clement's church. After her death, it is said she was buried in that same place. The Lord Pope always turns aside from the street and it is believed by many that this is done because of abhorrence of the event. Nor is she placed on the list of the holy pontiffs, both because of her female sex and on account of the foulness of the matter.

That’s right: not just a female Pope, but a pregnant female Pope!

In some versions, the crowd, seeing the Pope start to give birth, decides that this is not, in fact, a miracle revealed (related to the Immaculate Conception, perhaps), but instead the revelation of a hoax. So they get Old Testament on her, and stone her to death.

Then Benedict 3 takes over as Pope, and all is well, and all manner of things are well.

That version above is from a translation of Martin Polonus' Chronicon Pontificum et Imperatum.

No contemporary of Leo 4 or Benedict 3 wrote down anything about a Pope Joan, but by the 14th Century, the story is accepted as truth by just about everyone in Europe. Especially Protestant propagandists, who use the idea of Pope Joan as evidence of the depravity of the Church of Rome.

Some people think that Theodora and Marozia are one basis for the legend. Details of Medieval Popes being inspected to be sure they had testicles are debated hotly, though many Popes did apparently sit on an ancient Roman chair (that may have been a bidet), which had a hole in it through which testicles could be checked, if necessary. In other words, the details sound plausible, so many people did and do believe that she existed.

Even though she probably didn’t.