Yes, “Elle est prise” might mean “she is busy”, but depending on the context, it could also mean that she is in a relationship.
- “Est-ce qu’elle t’appellera demain?”
- Will she call you tomorrow?
- “Je ne sais pas; elle est prise par son travail.”
- I do not know; she got caught up with her work.
- “Tu veux sortir avec elle?”
- Do you want to ask her out?
“Oui, mais elle est prise.”
- Yes, but she is taken.
I hope that makes sense!