I don’t know that it really matters if she’s lying. Maybe she’s just hoping you’ll take the hint and give her some space. Some people are just not assertive in asking for what they need. You have the opportunity here to be a good friend and use your perceptive observations that she is struggling to help her by just giving her some space. If that’s what she needs, she’ll appreciate your ability to “read” her and maybe down the road when she’s in a better place she’ll be able to initiate some interaction and you’ll know at that point that she’s ready. If she doesn’t do so, then part of what she’s basically “saying” through her actions here is that she’s not interested in the friendship to the extent that you are.

Of course we would like people to be more direct, but for someone who may be in a really difficult personal space, it might just be beyond her an ability at this point.

Sometimes we just have to “read the room” and disengage from a situation rather than pushing.