The Richmond branch of NatWest is on a list of branches earmarked for closure.
The branch is one 179 across the country set to be shut, along with 62 Royal Bank of Scotland banks.
RBS, which owns the banks and is 71%-owned by the taxpayer, said it would try to ensure compulsory redundancies were “kept to an absolute minimum”.
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The bank said use of its branches by customers had fallen 40% since 2014, but the Unite union, which represents bank staff, called the cuts “savage”.
An RBS spokesperson said: “More and more of our customers are choosing to do their everyday banking online or on mobile.
“Since 2014 the number of customers using our branches across the UK has fallen by 40% and mobile transactions have increased by 73% over the same period. Over 5 million customers now use our mobile banking app and one in five only bank with us digitally.
“We realise this is difficult news for our colleagues and we are doing everything we can to support those affected.”
Earlier this year, Yorkshire Bank closed in Richmond, while HSBC shut its branch in Leyburn.