While banks can’t literally print their own money in a system with a central bank, they can increase the money supply. In a system of fiat currency, banks’ monetary base (i.e., what is actually in the “vaults”) is made up of money backed by the central bank. However, when banks make loans above their reserve (which is pretty much always), it adds to the money supply, specifically what economists call “M2” and “M3” (depending on the type of loan), which are considered less “liquid” than the monetary base. Thus, lending can (but not necessarily will) cause inflation.
********* wrong Todd
Yes, when banks loan out a fraction of deposits, they increase the money supply.