ok, the longer answer is -- that's incorrect. The Baptists can date themselves back to John Smyth in Amsterdam in 1609. Even prior to that, in 1606, John Smyth, a Fellow of Christ’s College, Cambridge, had broken his ties with the Church of England. Reared in the Church of England, he became "Puritan, Separatist, and then a Baptist Separatist," and ended his days working with the Mennonites. He then separated from the aptly named group, the Separatists (their descendents are the Congregationalists) in 1611 (sola scriptura, sola intepretura)
Since he wanted to join the Mennonites (Anabaptists), the term his followers took for themselves was BAptists.
In 1624, the Baptists pronounced anathema against the Anabaptists (more splitting)
So, as you see, the Baptists split from Puritans who split from Anglicans who split from The Church.