Did the Xbox 360 Have Cordless Controllers?

The Xbox 360 used wireless controllers (2.4 GHz RF) that required a USB wireless receiver (built into the console on "Slim"/"E" models, external dongle for original "Fat" models).

While technically "cordless" in operation (no wire to the console), they are classified as wireless, not "cordless" (a term often reserved for IR or early RF toys). They run on AA batteries or rechargeable battery packs.