However, even inflation will not be able to save the banking sector.

Do you have any evidence for this?

If I have money that I don't need right now, I may deposit it in a bank account against the promise of a fixed interest rate.

And that interest rate will be - at least in the countries where I have accounts - below the rate of inflation.

Thus, leaving the money in the bank promises exponential growth.

In nominal terms only.

In other words, your number will get bigger, but the value of that number will shrink, thanks to inflation.

Evidently, the bank must be able to reinvest my money in some business that offers a higher interest rate, i.e., faster exponential growth.

Sure - it could invest it in production that gets a zero real rate of return, which is the same as a nominal rate of return equal to inflation. Since it's paying you less than inflation, it earns money.

The only way how this can be offset is by accepting an inflation rate that equals the interest rate that the bank is paying for its investments, but in that case, the customer has no longer any incentive for depositing his money in the bank.

Would you rather lose 4%/yr or 2%/yr? Some interest is better than none.

You may believe nobody would put money in a bank at an interest rate lower than the inflation rate, but millions of people in the real world are in exactly that position right at this very moment in both Europe and the US.

i.e., your argument is based on an assumption, and that assumption doesn't fit reality. Now would be a good time to reassess your argument.

EDIT: just to be clear, your error here is in confusing nominal and real rates of return. If you get a zero nominal rate of interest there's no incentive to give your money to a bank, since you could just stuff the money under your mattress. By contrast, a zero real rate of interest is still valuable, since it corresponds to a (nominal) rate of interest equal to the rate of inflation.