According to Merriam-Webster, to undermine means "to subvert or weaken insidiously or secretly." This secrecy makes dealing with being undermined difficult. Often times, if a person is undermining you, she's doing so behind your back rather than to your face. To deal with the negative effects this person might have on your work, social or family life, you'll have to be the bigger person and confront her calmly but assertively.
Avoid direct confrontation with this person unless absolutely necessary. Most people who take time out of their day to undermine someone else have little effect on other people's opinions. Maintain good relationships with the people to whom this person is falsely undermining you.
Write down a script for the conversation you wish to have with this person if the undermining begins to affect an important aspect of your life. Write down details of what you'd like to say to this person. Separate out things you might want to say that sound too angry or sharp from the things that will lead you to a productive resolution. This will help you get your mind ready to have the talk.
Ask the person if you can discuss something privately with her at a future date. This prepares her for the conversation as she can begin thinking about what you might bring up. Catching someone off-guard with a personal conversation prepares her to react aggressively in defence.
Ask the person undermining you if you can have a third party present if you feel the conversation might get out of hand. This third party should have a general idea of what's going on. For example your third party could be a mutual friend if this is a social relationship, or a co-worker who's been involved in on the gossip if this is a problem at work.
Ask to present your side in its entirety before the other person starts talking. Set the rules of the conversation before beginning to talk so that you aren't interrupted. Tell the other person you'll gladly listen to her without interrupting after you're done.
Give the details of your personal feelings, as well as concrete examples from real-life situations to back them up. For example, if being undermined at work cost you membership on a team project, present this fact, then say honestly how that made you feel.
Listen closely as the other person speaks. Listen to her actual words, as well as the implied messages that might not be as obvious. For example, you may discover through confused speech and frantic body language that this person is simply envious of you.
Negotiate with the person. Set concrete terms and conditions. This might be an agreement to ask for separate work assignments or stop occupying the same social spaces, or it might be an agreement to stop talking about each other unless the other person is present.