At approximately 40:30 a corpse in the background can be seen breathing.
This episode continues the fallacy that every Pope is Italian. The current Pope, Francis, is Argentinian. The last Italian Pope was Pope John Paul I, who only reigned for a month in 1978 before his death.
Cardinal Angelo says that the Doctor is personally recommended in a letter written by Pope Benedict IX. Given that Benedict IX was deposed and during his reign broke his vows of celibacy and eventually sold the Papacy (the sin of simony) it is highly unlikely that a recommendation from him would be viewed favourably.
There is also nothing to suggest that Benedict IX was a woman, as the Doctor claims.
There is also nothing to suggest that Benedict IX was a woman, as the Doctor claims.
When the Doctor is talking to Cardinal Angelo about the suicides, there is a brief flashback of an Italian newspaper with the headline "Suicidi inspiegabili in tutto il paese" (Inexplicable suicides across the country). There is a photo of Italian police officers on the paper. Other flashes show scenes of the investigation and a gun being dusted for prints. Yet the evidence bag that a gun is put into (seen in a later flash) has English writing on it.
It is well established that the Tardis can translate any language, written or spoken into the language of the observer/listener, however the when the Pope speaks, he needs one of the Cardinals to translate for him.
Rafando mentioned that his device will deactivate both hearts and all three brain stems (plural) of a Time Lord. However, when Missy wanted to meet Clara in The Magician's Apprentice (2015) she asks for, "Eight snipers -- three for each heart and two for the brain stem" (singular).
When Bill and Nardole go through the door into the Pentagon, they're asked if they have clearance for "floor three," which is a very English way of saying "the third floor." Very few, if any people in the United States say "floor three" to refer to the third floor.