No, but not because they are a clone! Nobody has a spirit, in the traditional sense of the idea. See the answers to the following questions:
- What is the difference between soul and spirit? Do animals have a soul?
- How is “soul” used in the Bible?
- Does our soul or spirit go to heaven?
- Where do we go after we die?
- Where did Jesus’ soul or mind go when he died?
- Why did Adam and Eve need immortal souls and when did they get them?