This is a bit tricky but if you don't have to deal with date formats and locale it may work for you.
Place the UIDatePicker inside a UIView and align all sides to the containing view's edges. Then change the right side constraint from 0 to -100 (you may have to find a different number depending on the size you want to give it).
In the containing UIView's User Defined Runtime Attributes add a key named clipsToBounds, make it a Boolean and make sure it is checked.
This will hide the edge of the Date Picker that is aligned beyond the UIView's right side.
Make sure you set a minimum and maximum date for January 1st to December 31st of a specific year (so the user doesn't accidentally go outside that year and you don't have to deal with multiple years).