In a book "The mystery of the hanging gardens of Babylon", author Stephanie Dalley argues that there were such gardens, associated with a royal palace, and using Archimedean screws to water tiers of plants, but that they were located in Assyria (Nineveh), and built by Sennacherib around 700 BCE.

Here is a quick review of the aforementioned book by Dalley.

For an even quicker overview: The first extant textual references to the Hanging Gardens come from Greek authors in the 1st century BCE; several hundred years after the fall of the last Assyrian Empire, and much later from the height of Babylon. There's no archaeological evidence for significant water-works at Babylon itself, but Dalley has shown that "Babylon" was used in later texts as a name for multiple important cities across Mesopotamia. Her theory is that the Hanging Gardens were actually located at Nineveh in the Neo-Assyrian period, based on an inscription by Sennacherib describing what can be interpreted as a water screw, Archimedes-style.

I think I've heard claims of some ruins at Nineveh that could corroborate this, but it was a while ago and I can't recall precisely, I might be conflating some different things.

I'm planning on buying Dalley's book soon, I might post again with more info in a few weeks.