> This is a complicated question that will start a flamewar, but basically, any bank in the world can create dollars out of thin air given adequate collateral. Read about "Eurodollars".
This is true, but it's actually sort of a specious point. What a eurodollar deposit "really" is, is just a short contract. Your European bank takes your exchange and promises to give you back dollars when you ask for them. That they may not have them right now is just a technical detail. But the value of the dollar is still set by what the market is willing to pay for it. The bank going "short USD" is a risk it takes, not an inflationary pressure per se.
So... sure. Foreign banks can create dollars just like brokerages can "create stock shares"[1]. Except, they really can't. They're just playing accounting tricks.
[1] Or, more relevantly perhaps, how one crypto exchange can create wrapped tokens representing foreign coins. This capability was part of the exploit chain that led to the DeFi hack last month.