Broken water pump belt

I was driving at around 15 to 20 degrees Fahrenheit when I got a solid red light on left side of my gauges, and I did not know what it was, so I kept driving. Then after about three miles I got a red blinking light on the right side of the gauges, I pulled over right away and found that the water pump belt was broken. A bunch of white smoke appeared, however the coolant was not lost, it was still full, but it was very hot.
My question is do you think it will drive again if I replace the belt? Or do you think it caused major engine damage due to the cold weather? Again it was probably just a three mile drive after the belt broke, I think but cannot be sure, it definitely was not more than two or three miles.
Best regards, Stephen