She can apply for an extension of her J1 visa.
In order to get the extension she'll need to speak with the responsible officer at the institution where she's been studying.
Extensions are granted if she still needs more time to complete her current project for which she was approved for the J1 initially. It won't be granted for an extension of a new project. But if she still needs more time on her current research or project, she may be able to get an extension of her visa.
She should meet with her responsible officer as soon as possible though, because the extension needs to be submitted before her current visa expires.
If she is granted the extension, this could afford her the time she needs to get her petition for political asylum submitted and approved.
She wants to avoid a situation where she ends up overstaying her visa in the US, because here is the problem she can run into:
- if she overstays her visa by 6 months, she will face a 5 year re-entry bar to the US
- if she overstays her visa by 12 months, she will face a 10 year re-entry bar to the US
Therefore she wants to make sure to either get an extension granted, or get her political asylum petition turned in, before the expiration of her visa.
As far as if she is living with you later on with an expired visa, no you won't face any liability or sanctions for that.
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