Government "shouldn't" have anything to do with it. But Central Banks (not the same as the banks you are referring to) have everything to do with it. Banks cannot lend money which they don't have. No, they can't just "create" money. Central banks do that. It is true though, that there is not enough currency in most banking systems for the actually cash to be completely withdrawn, as much of it is digital fantasy. However the central banks are the ones that put it into the "system".
Yes, banks CAN just create money. If someone gets an credit from the bank this credit isn't based on money the bank has accumulated from other sources. It is written to the account as book money. True is, that these credits must be backed by some money the banks have from other sources. How much is not that easy to answer. For europe and the US the current rules are based in the Basel III agreement:
It's hard to understand, the rules are complicated. But basically the credits are money which is created out of thin air and backed by around 3%-5% money the bank has in reserve. this reserve money can come from a credit the bank took from another bank.
That the central bank isn't creating money alone (they do too based on the same principles) is clear, if you see that the central banks don't know how much money is in existance but make estimations instead.
The central bank controls the actually printed money (M0) and some deposits (MB). Other than that it is out of direct control for the central bank. M2 includes the money you have in your banking account. In the graphic you see how much more M2 (green) is in comparison to MB (blue). So M2 is mostly money that is NOT created by the central bank.
The situation in europe is similar.