Factual error: In the sex scene, when Adam Driver pins Lady Gaga to the wall, the map of Milano on the wall says "Mailand." Now there are parts of Italy where they speak German, but Milan is not part of it, so it's highly unlikely that they have a German map of the town (which wouldn't be produced in the German-speaking parts either). Since it's in her father's office, the map cannot be from Gucci's mother (who happened to be of German origin).
Factual error: When Patrizia Reggiani signs cheques on her father's behalf, the bank that issues them is "Banca di Credito Colli Briani." Reasonably enough, is a completely made up name for a bank, but also bizarre because "Colli Briani" would be a toponym ("Briani Hills") but does not match any name in Italy - at most "Colli Briantei", which would be the Brianza region, not far from Milan where the story takes place. No other names of places or people in the rest of the movie are fictional. Moreover, the cheques don't have the number of the account they are associated with, and they are out of order. (00:03:40)
Factual error: When the Financial Police (Guardia di Finanza) arrives at Maurizio's Mansion, it should be in the late 80s, but they drive an Alfa Romeo 155. This car was produced from 1992 until 1998. (01:29:21)
Factual error: When Tom Ford reads The New York Times article about his successful show, his copy from Sunday March 13 1995 is a Frankenstein made out of different articles from different parts of the year that don't match the titles (the 'utility employees' article is actually about a single mother in a dire situation, published on December 26, the 'Soviet ship' article is from November 10). Also, there's the small detail that March 13 1995 was a Monday, not a Sunday. (02:16:35)
Continuity mistake: When Maurizio corrects Patrizia saying the restaurant was at the second floor of the Tour Eiffel, Lady Gaga goes from both hands joined to one hand up in the air in the middle of a gesture. An even more blatant instance later when he lambasts her saying they wouldn't tell her if they did mind her gab. (01:36:05)