DIAGNOSIS That's not a defect, it's a feature. Dyson uses a thick, rubberized cord on the Airwrap that's designed to be heat-resistant and durable. It's notoriously stiff and has a memory—it will hold any kink or bend you put in it.
FIX To store it, don't fight it by trying to make tight coils. Gently loop it in large, loose circles. The best practice is to use the built-in wrap cleats on the storage case. Wrap the cord around those cleats in a figure-eight pattern, which minimizes sharp bends. Over time, it might soften slightly, but it will never be a flexible cord.
IF THAT DIDN'T WORK If the cord is stiff to the point of being cracked, or if the plug feels loose in the socket, that's a different issue. Inspect for physical damage. A damaged cord is a safety hazard and needs replacement from Dyson.
HEADS UP Never wrap the cord tightly around the machine handle when it's hot, as the heat can further set the bends and potentially damage the insulation over time.