Miss Kay, the family matriarch, was rushed to the hospital Tuesday after being attacked at home by her dog, Bobo ... according to Phil and Jase Robertson.

Phil shared the dog attack story on their podcast, "Unashamed" ... and he says he was awoken in the middle of the night by a bleeding Kay, who had a rag over her mouth as she told him she needed to get to a hospital STAT.

Phil says the dog split Kay's top lip in two, with the gash about a quarter-inch deep. He says she was bleeding all over the place and there was a notable chunk missing.

According to the family, Miss Kay was going in for a goodnight kiss when Bobo snapped back ... Bobo was already sleeping and she woke him up, and the pooch is not getting the blame.