No, not theoretically.

FATCA does not apply once you are no longer a US taxpayer. Moreover, the FATCA regime has been copied by so many other countries (including the EU) that there is now a global version of FATCA called CRS (and the procedural system for implementing CRS is known as AEOI).

FATCA is now considered the least burdensome implementation of CRS, which has been implemented by all OECD countries, China, India, and Brazil.

If a bank is turning you down because you answered "YES" to the FATCA question, the problem isn't FATCA. The bank is trying to avoid an audit and it's a huge red flag not to do business with them.