The Welsh Liberal Democrat leader said the online petition is in response to a spate of bank closures across the county.
Banking giant HSBC this month announced it will close its branches at Rhayader and Builth Wells in March next year.
The announcement follows the closure in November of the Barclays bank in Llanwrtyd Well and the Natwest branch in Crickhowell.
Last year HSBC closed its branch in Ystradgynlais and the Natwest closed its branch in Talgarth.
Ms Williams said: "I am absolutely appalled that HSBC are deserting two vibrant rural communities. I have written to the bank’s chief executive at the bank requesting a meeting so I can tell them just how disgusted I am by this news.
"Since May 2015 we have already lost two local bank branches in Brecon and Radnorshire, in Llanwrtyd and Crickhowell and I know many other rural communities across Wales are suffering too. It is time the UK Government did more to stop the closure of rural banks.
"Given that HSBC’s pre-tax profit in the first six months of 2015 was £8.7bn, it is quite difficult to believe that these two branches are unaffordable. It seems the slogan ‘the worlds local bank’ are just meaningless words. HSBC needs to start putting its rural customers ahead of profits."
Brecon and Radnorshire MP Chris Davies said he hopes to meet with the bank to discuss the branches.The Conservative said: "The decision by HSBC to close their branches in Rhayader and Builth Wells is deeply disappointing. These banks are central to community and business life and I will be robust in my opposition to HSBC's decision. I want HSBC to recognise that in rural communities such as Brecon and Radnorshire it is not as easy as popping to the next town to go to the bank, when the next town can be many miles away.
The bank has said it recognised closures are "unsettling" for the local community but use of the branches has fallen "significantly". It said there has been a rise in online and telephone banking.
It said customers will be able to use Post Offices to pay in and withdraw cash. Other banks have also made arrangements for customers to use Post Offices when branches have closed.
The petition has attracted more than 170 signatures in its first 24 hours. It can be found on the AM’s website http://www.kirstywilliams.org.uk/save_our_rural_banks






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