No
So far as I'm aware there is no law explicitly against an "encrypted ledger" but retail banks have legal obligations that can only be met by awareness of activity in user accounts. They entail being able to see user accounts at transaction level.
E.g. rules relating to
money laundering
terrorism financing
serious crime financing
other activity considered suspicious in the jurisdiction
reporting on certain customers (persons or entities) under the Common Reporting Standard
risk, exposure
how much money the user had that is 'guaranteed' to be paid in the event of a crash
In at least the past decade the legal trends are towards greater transparency and traceability, less anonymity and financial privacy, and directly opening financial records to law enforcement authorities. And now we have Open Banking in the UK, EU and I think developing in the USA where the biggest banks are (going to be) obliged to facilitate licensed startups direct access to user accounts.
That's all aside from the bank being able to meet its obligations per the contract with the user and check the user is meeting his contractual obligations too. E.g.
is the user attempting to withdraw money over a threshold that he may not exceed?
did the user exceed their authorised overdraft (therefore can be charged higher interest and/or penalty)
calculating interest due either way (to the user or bank's benefit)
whether the account qualifies for positive or negative treatment due to frequency of withdrawals (e.g. supersaver account attracting higher interest may allow only five withdrawals a year)
investigating user complaints (e.g. fraudulent transactions or mistakes)
fixing mistakes
processing directcredits and debits, standing orders, payments
processing penalties
freezing or blocking certain amounts
(Indeed how could such a business operate as a retail bank?)