L.C.
I agree with others that you need to tell her the truth. Making up stories (such as the "owie story or no more milk) only backfires later in eroding trust in this most precious relationship.
Difficulties like this are wonderful teaching/ learning opportunities. As in any tough situation the best way is to listen and acknowledge her frustration and anguish while remaining firm about how things are. You show compassion and she gets to deal with reality.
I also agree that it's a great idea to substitute something positive. I weaned my daughter from night-time nursing when she was three. I transformed the nursing times to rocking and cuddling times. So we still had that closeness and maybe I would sing lullabies but stayed firm on the no nursing once we had established it.