If you keep doing that thing even after that person is gone, you carry on without her. Example: "If anything ever happens to me, I want you to carry on without me." Another example: "It is true that our best player has broken his leg, but we will carry on without him."

To carry on without someone is to continue doing that thing after that person is gone. Example: "I'm done playing for today. Carry on without me." Another example: "Really, I need to go. You two carry on without me." Another example: "Her husband died, and she has carried on without him."

If you lose some thing but continue doing what you were doing, you carry on without that thing. Example: "Her shoe fell off in the middle of the race, but she carried on without it."