How to deal with failing a student?
Wait for two days, and if she doesn't give me an exact delivery date (which she probably won't), just give her a fail grade.


Given that you already agreed to this extension, this is the only option you have. Do this.

Honestly, you should not even have agreed to this extension. I understand that you were under stress when you agreed to it, but students making a scene will happen more often to you in the future, and at some point you will get inured to it.




  I feel very bad failing a student in her first big project,
  especially since if she fails she probably need to study one more
  semester to finish her studies.


This seems to be your real question. Unfortunately, there is really no way around this dilemma. If someone persistently does not deliver assignments and refuses all offers of help, failing them is the only option. Anything else will make a mockery of the entire system of higher education. (And be sure that other students will notice.)

Look at it this way: if you pass her, and she gets a degree, what will her employer do with her if she continues this way? They will fire her, and quickly so.



It appears that this students needs some serious help and counselling. It appears like you already did all you could do in this regard. I would recommend that you wait for two days, fail her if she does not hand in the date, and write up an explanation like you did here. Send this explanation to her in writing. Explain that it appears she is not ready to take a course like yours. Direct her to any counselling resources available at your institution. Keep a copy of this explanation for your file, in case she escalates this to your department head/dean/whatever.

Then do not enter into any further discussions with her, unless she retakes your course.