The answer to this question depends on the type of surrogacy. If it’s traditional, then yes. The baby will have the surrogate mother’s DNA because her egg was the one fertilized to create that baby.  Meanwhile, if it’s only gestational surrogacy, then no. The baby will have only the DNA of the egg and sperm used to create it. They could be from the intended parent or from a sperm or egg donor. The surrogate mother will only be a gestational carrier.