Hi @onalex
Before the three months have ended, she can apply for an extension of the Schengen visa with the Ausländerbehörde. However, an extension is only allowed in very exceptional cases. In my view, a long waiting time at the embassy is not sufficient for an extension (although of course very understandably frustrating). Here you can see the information from the authorities in Berlin about what kind of cases would be eligible for an extension (in English via online translation):
https://service-berlin-de.translate.goog/dienstleistung/324785/?_x_tr_sl=de&_x_tr_tl=en&_x_tr_hl=en&_x_tr_pto=wapp
If this is refused or if the three months have passed, she is obliged to leave the country. She can then apply for a Duldung but again here I do not see sufficient grounds for her being issued with one. A Duldung can be issued for humanitarian reasons but a long waiting time for a visa is not such a reason. I imagine the application would be rejected and she would be obliged to leave the country within a certain deadline.
I do not know what influence such a rejection (i.e. to a visa extension or to a Duldung) would have on her subsequent long-term visa application. At the bottom of the application form the applicant is required to make a declaration that they have not been expelled from Germany or had an application for a residence permit rejected:
https://videx.diplo.de/videx/visum-erfassung/#/videx-langfristiger-aufenthalt
While this would not exactly apply in her case (as she would not have applied for a residence permit), if you consider such a course of action you should first consult with a lawyer for immigration law. Such a case would probably have to be argued in court and as I wrote above I do not think it likely for such a case to be successful. But I am not a lawyer and so not gauge this.
Best,
Éanna