The couple shared the news on Monday ... announcing Kaius Green Jeter was born on May 5.

"Welcome to the world lil man!!!" the Yankees legend said on Instagram.

The new arrival comes as a surprise to pretty much everyone -- the two kept the whole thing a secret throughout the pregnancy, which the Sports Illustrated swimsuit cover model has done in the past.

Derek and Hannah -- who got engaged in 2015 and tied the knot a year later -- also have three other children together ... Bella (5), Story (4) and River (1).

It's an amazing time for the happy couple ... and we're sure Derek can't wait 'til Kaius gets old enough to play catch in the backyard.

Of course, the kid can be a quantum physicist if he wants ... but new Yankees shortstop coming in a couple decades?!?!