Coworker drives me mad

By the way, rationally I know a coworker cannot drive me mad. It is my thoughts.
I work as a project manager in a client facing role. Essentially I’m managing an internal team at my workplace, but am accountable to and I am the main point of contact for a client.
I work with a woman who frustrates me a lot – she tells work-related lies to me and the client (I try and call her out on this professionally when I can and when I have proof) and can never agree to a deadline. Because I am the client facing contact, I cop the brunt of their frustration when she misses deadlines or makes up stories.
I have tried a number of communication strategies to get her to work with me (setting false deadlines, offering to help as much as I can, etc) but now just avoid being on projects with her. How can I excel at my job when I feel I’m at the mercy of others’ actions?