that denmark negative interest rate is interesting. But it still doesn't quite answer the central issue - who is "giving up" money to lend? The guardian article (https://www.theguardian.com/money/2019aug/13/danish-bank-lau...) only mentions that it's banks passing on negative rates from institutional lenders in money markets.

Sorry, I do not quite understand your question.

Anyway, it is ultimately (when you get your head around it, that is...) relatively simple. These institutional investors have cash (well, not exactly cash, but "cash" as in overnight deposits) that they need to invest. some poart of that they want to invest in "safe" instruments, e.g. bonds/commercial papers issued by banks. And it happens to be that they are willing to do that even if they are paid negative interest rate[1]. So the bank pays for its funding say -1.0%. Then the bank can lend the money forward with -0.5% and still make money.

[1] Why do they want to do that? If you have a decent amount of money, say millions, and you want to save that for a while to purchase something say next month, you have not that many good alternatives to bank deposits/commercial papers/bonds. Actual cash is surprisingly cumbersome and expensive. Stocks are risky, you may lose quite a bit within one month. Government bonds pay even less. so if your bank says yep, we take your money for a month and pay you back a bit less in a month, you are likely going to swallow it.

Maybe it is worthwhile to note that there is no money[1] that would not be someones loan from someone. Your moeny at the bank account? It literally only is "money" that you have lent to your bank. Cash? A bearer note from the central bank, i.e. a token that says that you are owed by the central bank.

[1] As per current typical definitions of money. If you consider gold or e.g. cryptocurrencies as money, then the statement does not hold.