This is an extraordinarily deep question. The short answer is: copper ingots in Assyria around 1200 BC or definitely coins in the Mediterranean around 700 BC are our best guesses, but there is strong evidence that we are missing many earlier examples in prehistory.

To understand why this is hard to figure, it is worth considering what currency is and why it is important. As first stated by Jevons, the Double Coincidence of Wants Problem is the feature that makes any economy relying on barter alone inefficient. Using some exchangeable store of value, which all parties agree is valuable because they believe all other parties think it is valuable, solves this problem. Simply put, the idea of money is a fiction that makes us all better off. It allows for greater specialization in the economy, as people can focus on producing things directly demanded by relatively few others. (Whereas before you needed to provide something of value to enough others that you could always barter for food.)

There is considerable evidence that people stumble upon the money delusion independently and often. There are examples of Pacific Islanders exchanging large stone wheels, native-Americans and Africans using shells and beads, et cetera. There may be a trick of human psychology that allows us to ascribe “real” abstract value to shiny things and trinkets, and that abstraction often accidentally turns into something useful, even vital, to civilization. And note, lots of these forms of money are not actually very durable. So we would have no archeological record, and even then wouldn’t understand it if we found it without cultural context.

So while cuneiform records of copper ingot exchange in the 12 Century BC might be the first time we have a clear record of money being used as money, it is easy for this to be as much about our ability to observe as the history of human behavior. We do not have proof that complex economic activity cannot occur without money, but there is a strong circumstantial case to be made. Look at artifacts of household goods from the ruins of Akrotiri, and you will see signs of professional specialization that would be very hard to achieve in a barter environment. (It is possible slavery was used as the labor specialization process instead, but this drives us into a complex theoretical debate. Suffice to say, a surprisingly large, richly appointed city on a small island with an economy built on trade would be difficult to create and sustain using slavery and caste alone to specialize labor.)

My background is in economics, and I have read extensively on this topic. But I can’t point to a perfect work. A popular volume on this is David Graeber’s Debt which addresses some of these issues. I don’t actually like many of his conclusions, but he’s from an intellectual tradition with which I have grievances. My view of the scholarship of this question is: it is hard to publish a book coming to the conclusion “we haven’t got a clue”. But I honestly think, when it comes to the beginning of currency and money, we haven’t got a clue. If humans were trading shamanistic relics as fungible units of exchange 15,000 years ago, it would be totally consistent with our other observations and nearly impossible to observe or refute through existing methods.