Thank you for asking this important question. I can answer in the supreme confidence of knowing that my answer will constitute as definitive an answer as could be expected considering the complexities that this particular answer must necessarily address. For instance, the answer to this question must certainly descend to a pre-historic level, to a time pre-dating the written word, where we may rely on rumor and innuendo alone as passed on from generation to generation in odes and songs of bards from around the campfires and within the long houses of ancient times. However reliable, on that level we have it to a reasonable degree of certainty that the first banking institution known to mankind was the locally administered and meagerly capitalized, yet time-honored Bank of Mom and Dad.
It will come as no surprise that this stalwart concern (though practically devoid of investors) was not a depository, but a lending institution only. And it will also come as no surprise, I’m sure, that the Bank eventually failed, which failure was due largely to its foundering credit portfolio, and had to be bailed out by the Bank of the Rich Uncle, which Bank continues to this day though having changed proprietary uncles over the ensuing eons, and whose reigning proprietary is one well known in certain circles associated with asset recovery programs here in the United States.
Of course, on a more contemporary level, and well within the realm of recorded history, though it was, in deference to that stated above, assuredly not the first-ever bank in the history of the world, but presumably one of the earlier regional banks established in the history of Western Civilization, a royal edict was published in France on the 5th of May in 1716 authorizing the establishment of a bank under the name of Law and Company, whose proprietor, John Law, was actually Scottish, not French, and the notes of which would be received in payment of the taxes. This was during the reign of the extravagant Louis the Fourteenth. Branches of this Bank were almost simultaneously established at Lyons, Rochelle, Tours, Amiens, and Orleans. Letters patent were issued to incorporate the Company in August 1717. This Bank’s notes were not backed under the gold standard as are ours today, but were backed by real estate: that is, the value of all of the lands in France at the time. This was perhaps the first-ever land bank.
I hope this helps you in your pursuit of information regarding the banks of the world.