The TSB branch in Richmond is among 164 across the UK set to close next year.
There has been a bank on the premises in Frenchgate for 170 years, with the branch starting life in 1851 as a York County Savings Bank.
TSB has blamed “a significant shift in customer behaviour” as more customers bank online on the decision to make the closures
The 164 branches set to shut next year is in addition to the 82 branches it said it would close in November, when it set out plans to save £100m by 2022.
A list of the affected branches and their closure dates has now been published after staff were informed.
Union Unite described the move as “a dark day for the finance sector”.
The bank said the closures were not an “easy decision” and had been accelerated by the pandemic.
From the end of next year TSB will have 290 branches – down from 475 currently.