I often see a range of answers to this and I'm not sure what to think now. I've seen everything from the Akkadians to the Neo-Assyrians to the Imperial Romans.
A recent podcast I heard even claimed one of the Sumerian rulers (of Lagash I think) had a standing army. He was the one that had unbroken victories for decades and beat a coalition of Sumerians, Arameans and Elamites IIRC. His name escapes me but I hope someone knows who I'm talking about.
How would you define true standing army?
Virtually any monarchy in history would have some sort of professional guard retinue. Does that count as a standing army?
The Sacred Band of Thebes was a standing group of 300 soldiers, among a legion of regular phalanx troops. Do they count?
The Mongols probably satisfy the requirements of being a standing army, but nomadic armies don't usually seem to count as standing armies for whatever reason.
The concept is too vague to answer accurately. Easier to ask what's the first modern standing army.