TL;DR: The DropDownList widget cannot be closed by clicking the arrow-icon.

The long story:

After the DropDownList has been opened once, it closes and reopens every time the user clicks the arrow-icon. Closing the DropDownList still works when clicking anywhere on the dropDown except the arrow-icon (or rather the button around it) or anywhere else but the dropDown. This DOJO demonstrates the problem. Animations have been disabled to allow easier reproduction (with animations on, the problem appears only occasionally, probably because of timing issues).

If you look closely, you will notice that the dropDown closes when the mouse button is pressed and reopens when the mouse button is released (this is when the click event is fired). Therefore, the problem can be (temporarily) mitigated by suppressing the mousedown event on the dropDown's wrapper element.