A new coworker seems to want my position within the company and has been constantly trying to undermine me in meetings and in front of my direct reports. She constantly tries to make up things that are not true to make me seem unqualified for my job. She is more outgoing than me, so people may like her better. How do you think I should approach the situation to be able to deal with her on daily basis and not get aggravated? I need to beat her at her own game, but in a more sophisticated way.
I am going to assume that the facts are your side and that your new office mate is really undermining you and is trying to take your job. So here is some tactical advice:
Establish a paper trail of your work efforts. To me that means git logs, emails, minutes of meetings, your reports, etc.
Talk to HR about this. If you are too afraid of disclosing to them what she is doing to you, maybe ask for an assessment of your own performance.
Confirm if she is really undermining your efforts. You do not want to come across to everyone as insecure if you start confronting her about this.
Focus on the quality of your work. That way you'll have more documentation on your side.
Setup your workplace such that your efforts would be advertised by your colleagues if you make a significant positive contribution, such as closing a deal.
Avoid teaming up with her whenever you can. Those types are likely to expect you to pull all the legwork and take all the credit, or at least make it appear to the everyone that she did her part when actually. She didn't.
To me it is really important to document your work efforts because it'll be more difficult for her to prove that your work is useless.
She is more outgoing than me, so people may like her better.
This is not necessarily true. Others in the office will notice her behaviour, even if they don't know you. If all they hear from her are negative comments they might even take a disliking to her.
I need to beat her at her own game, but in a more sophisticated way.
I don't think you need to "beat her". Your work should speak for itself. The boss or manager who hired you will know what your skills and qualifications are, and as long as you are doing your job to a satisfactory level, the results are the best testimony you have. Your coworkers will recognise this also.