Santander has said "key services are recovering" following customers' problems accessing accounts online and withdrawing cash from its ATMs.
The bank, which has 14 million active UK customers, said earlier that the problems affected branch, telephone, online, app and card services.
In an updated statement, the bank said it was working on a "full recovery by the end of the day".
It apologised and insisted: "No-one will be out of pocket."
From Monday, the bank said it will post details on its website and social media channels about how impacted customers can make a claim.
One customer told the BBC he had lost a day's wages because he could not buy petrol to travel to his job as a CCTV operator.
"It's been an absolute nightmare," said Marcus Evans, from Aberdare in south Wales. "Nothing is working. It's ridiculous.
"I get my military pension and my wages paid into a Co-operative savings account and I transfer money on a weekly basis into my Santander account.
"I transferred money this morning. It's left my Co-op account but it hasn't arrived in my Santander account.
"We're left with very little food. It's disgusting. They need to compensate customers for this."
'I'm leaving the bank'
Another customer, Stephen Curtis, 58, from Gloucester, said he was trying to buy a new car when his card was rejected despite having enough funds in his account.
"It was so embarrassing," he said. "This is unforgivable. I work in IT, I know about systems, I know about backups and upgrades. To have this happen, I would get the sack if it happened in my job.
"I'm definitely leaving the bank, and my wife's with the bank as well. We're on a tight schedule anyway, we're going on holiday Monday and we need the car for the holiday."
A third customer, Jason Crossland, from Rotherham, told the BBC he was stuck in an outlet shopping centre with his children, unable to buy enough petrol to get home.
"I'm just trying to keep the kids occupied and my mother's having to drive from Rotherham about 40 miles away, he said. "I've been online - nothing. I can't get anything.
"I've been having issues as soon as I started trying to use the card about 10.15."
The head of money at consumer magazine Which? said many customers will be stressed, "with people reporting that they have been unable to make online payments or in some cases purchase food in their local supermarket".
"Customers can incur fines, penalties and fees when they're not able to access their finances, so the bank must offer compensation to all those who have been impacted in this way," added Gareth Shaw.
The UK's financial watchdog, the Financial Conduct Authority, said it was "aware of the issue and in contact with the firm to ensure it resolves it as soon as possible".
Earlier, Santander apologised on Twitter that a "technical problem" was affecting services and told customers they could "access cash from other banks' ATMs, at the Post Office and can get cashback where that's available".
A service status page on its website had said planned maintenance was due to be performed on the Santander mobile banking app overnight on Friday. It is not known whether this prompted Saturday's problems.