Swift UIDatePicker for text field input - how to make UIDatePicker disappear?

You could try something as gestureRecognizer to make the UIDatePicker disappear:

@objc func viewTapped(gestureRecognizer: UITapGestureRecognizer) {
   view.endEditing(true)    
}

This is what I used in my app so when the user taps on anywhere on the screen the UIDatePicker goes away!

You call resignFirstResponder on the text field:

@IBAction func DoneButton(sender: UIButton) {
    dateField.resignFirstResponder()
}

Also, it's not a great idea to have a method and a property with the same name - not sure about precedence there, but chances are it could be confusing at some point.