Diagnosis
The physical mechanism for the cold shot button has failed. It's a known point of wear. The button is designed to be held down for a burst of cold air, and when released, a spring should push it back up to re-engage the heat. That spring or the plastic latch has likely broken.
How to Fix It
There is no safe user-serviceable fix for this. The button assembly is integrated into the body of the dryer. Attempting to pry it open will almost certainly cause further damage and void any remaining warranty.
If That Didn't Work
This isn't an 'if' scenario—you need to contact Dyson support. This is a recognized hardware fault. They will likely offer a repair or replacement, especially if the dryer is still under its warranty period.
Heads Up
Don't try to wedge something under the button to keep it up. You could force it into a position that causes a short circuit or prevents the thermal protection from working correctly.