The Huge Entity reports that the world's "oldest baby" was 17 months, 11 days in the making.
The result of the longest recorded human gestation, Baby August was the oldest newborn ever. [Link]The mother was Anissa August, M. D., according to this page.
One of the residents to deliver a healthy baby boy last year was Dr. August. Her pregnancy is also of note because it was the longest recorded human gestation on record. Dr. August was pregnant for a staggering 17 months 11 days.Sadly, the other pages on the site reveal that this was a joke.
I did find a 1945 Time article about a reported 375 day pregnancy. Some mothers need to learn when to let go.