First Mushroom Bank
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The Super Mario Bros. Super Show!
The first appearance of a First Mushroom Bank was during The Super Mario Bros. Super Show! episode "The Unzappables". This branch was located in Crime Land and was robbed by Al Koopone, Mouser, Koopa Troopa and Tryclyde. After Al Koopone blew up the vault door with a Bob-omb, Mario and his friends attempted to intercept his attack, but they failed to stop the Koopas because they were wearing their new Unzappable Hats.
The Adventures of Super Mario Bros. 3
The next appearance of a First Mushroom Bank was during The Adventures of Super Mario Bros. 3 episode "Crimes R Us". After receiving a crash course in bank robbery from Crimewave Clyde, the Koopalings pulled a bank heist on the First Mushroom Bank, demanding for all of the people there to give them their valuables. Bully also demanded for one of the bankers to toss all of the money from a safe into a sack.
Fortunately for the bank, Mario and Luigi showed up at the bank in time to make the Koopalings retreat, but not without Raccoon Mario getting tied up in a stolen fur by Kootie Pie, and Fire Luigi being doused with a fire extinguisher by Bully.