According to the best available sources, Col William Ball of Millenbeck and Hannah Atherold Ball had four children, Richard, William II, Joseph, and Hannah. A letter from Col. James Ball, Jr., “of Bewdley” dated 11 Sep 1789, to his nephew Col. Burgess Ball, is one of the oldest sources used to support certain claims about the family of William Ball of Millenbeck. Even it, however, contains some errors, but it is presumably accurate concerning the names of his children. In it he states, "It appears from a memorandum of Joseph Ball Esq., that [Col William Ball] married in London the 2nd day of July, 1638, a Miss Hannah Atherall: by whom he had Richard, who it is supposed died an infant, as he is not mentioned in his father's will; William born the 2nd of June 1641; Joseph born the 25th of May, 1649; and Hannah born about the middle of March 1650; but whether they were born in England or Virginia I cannot learn." (https://www.wikitree.com/wiki/Space:James_Ball_1789_Letter_to_Burge...)
The second main source, the Downman Family Bible, may be even older; although it's not clear when its entries were actually recorded, it was probably no earlier than 1776, the date Fanny Downman married Colonel James Ball Jr. of 'Bewdley'. It states that Col William Ball "married Hannah Atherold & died in 1680, leaving two sons William & Joseph & one daughter Hannah who married David Fox" (https://maryballwashington.com/III.downmanBible.pdf). Note that it erroneously omits their son Richard, but it also doesn't name a daughter Elizabeth.
As noted by Col James Ball Jr., the will of Col William Ball, dated 5 Oct 1680, does not list son Richard; it only names sons Willam and Joseph and daughter Hannah. It also doesn't list a daughter named Elizabeth, who would have been alive in 1680. (https://www.wikitree.com/wiki/Ball-186#Last_Will_.26_Testament)
Despite the flaws in these and later sources, most of which are secondary and derivative, they're the best available evidence, and they do not list Col William Ball and his wife, Hannah, as having a daughter named Elizabeth. I believe Elizabeth Ann Pope should be detached as a daughter of Col William and Hannah Ball.