"Depending on the definition, the world's oldest bank is either Banca Monte dei Paschi di Siena or Berenberg Bank. Banca Monte dei Paschi di Siena was founded in its present form in 1624, but traces its history to a mount of piety founded in 1472. The Berenberg company was founded in 1590 and has operated continuously ever since with the same family as owners or major co-owners. Banca Monte dei Paschi di Siena is today a large Italian retail bank, while Berenberg Bank is primarily involved in investment banking and private banking for wealthy customers; in any event Berenberg Bank is the world's oldest merchant bank or investment bank. The world's oldest central bank is the Sveriges Riksbank, which was founded in 1668."

https://en.wikipedia.org/wiki/List_of_oldest_banks_in_continuous_operation#:~:text=Depending%20on%20the%20definition%2C%20the,of%20piety%20founded%20in%201472.

Knights Templar are said to have started the first banking system.    Freemasons are said to be the Templars under a new name.   But they are no longer in banking.

Spunky listed the oldest in the world. Bank of New York is the oldest in USA still existing. This post was edited by my2cents at June 6, 2022 2:01 PM MDT

I believe the first "bank" was either a cave or a hole in the ground.  Put it in and hope nobody saw you do it. 

(Need to hide your money?  Then come on down to Balthazar's treasure emporium.  Whether you have a little or a lot, we can hide your money where nobody will find it!  We guarantee that our hiding pieces are the lowest in town.  If you find someone offering a lower price, we’ll either kill them or match it!  So come on down... and tell them Balthazar sent you.)

The West bank of the Euphrates.