Thanks for the further details.
You will be able to achieve the desired by modifying the default ControlTemplate of the DateTimePicker in order to remove the VisualStateGroup named CommonStates and the borders named BackgroundVisual, MouseOverVisual, FocusVisual as well as the WatermarkTextBox named PART_DateTimeInput. This way will have just have the drop down button as required. For your convenience I have prepared a sample project which demonstrates the exact approach, note that I have used Implicit Styles
in order to easily extract and modify the ControlTemplate.
Hope this will work for you.
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