Not in my experience. I live in Italy for some of the year and I can use my Switch, Mastercard and Visa cards at the local Casa di Risparmio (Savings Bank) in a town with a population of less than 2,000. I've travelled over much of Northern and Central Italy and never had a problem obtaining cash. Every bank on the street in Central Italy has an ATM.
Chiantishire , well even there despite being peculiar Northern bastards they have banks and ATMs.
While you're at it why not go and see the rest of Italy rather than the British owned part?
These are the sort of stupid banking rules which really pee me off. Imagine moving into a town with a population of 1999, and as soon as you move in you find you can't use the ATM's anymore.
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