Your text book is not very clear. B Could easily be "Mary is prettier than she is." Using the book's logic of "leaving out", then B is exactly correct.
In modern English, though, we tend to choose between the "correct" (in old grammar) subject pronoun with verb ( "than she is", "as he is", "as I am") or the now-standard and also correct, I think, object pronoun on its own; "than me", "than her", "than him".
Have a look at this link:
I should add that I have just looked online, and there are quite a lot of inaccurate sources on this, so your grammar book is not alone.
In English we do not have any final authority on grammar - no Academy or government department to announce what is true.
Most grammarians, I think, regard Merriam-Webster and the Oxford English Dictionary as being as close as we get to authoritative sources. perhaps adding The British Council, and the dictionaries from Longmans, Collins and Cambridge.