It seems unlikely that you have any living fourth cousins six times removed. That relationship means that your great-great-great-grandparents were her great-great-great-great-great-great-great-great-great-grandparents, or the other way around.
On the other hand, generations can be very long (President John Tyler, born in 1790, still has living grandchildren), so it's not completely impossible.
[edit: this is a reply to Stack47. Forgot to click the quote post option.]