Straight answer

Is a different surname a problem when travelling with a child?

Usually fine, occasionally questioned, and worth being ready for. Border officers at some destinations, and airlines on occasion, may ask you to show the connection. Carry the child's birth certificate showing the link, and if you are travelling without the other parent, a short signed letter of consent from them with a contact number. It is rarely asked for; when it is asked for, having it saves your morning.