Cows milk protein in lactose
Yes, process lactose could still contain protein from the milk. It's equally likely that the casein could contain lactose. Whey is much more likely to contain lactose since the whey and lactose are in the water when the cheese is made.

It would be extremely rare for a baby to be lactose intolerant since lactose is the major sugar they consume. As the child ages, the ability to digest lactose decreases. I think I heard that 5% of Caucasians adults can still digest lactose. Only a few people develop lactose intolerance.