Students trying to negotiate away penalties for late submission of coursework
Adhere to the guidelines and syllabus you posted. If students can get around consequences of late submission by arguing, you have set a precedent, and they (and future generations of students) will argue again the next time. Don't go there. Consciously cultivate a reputation that pointless arguments don't work with you.

Next time, make it clear that "normal" IT problems like corrupted files or network lag is not your responsibility, and encourage students to upload their work sufficiently early so they won't run afoul of such problems. If possible, allow students to change their submissions, so they can upload whatever they have a week before the deadline, and keep on uploading polished versions, with the last successful submission before the deadline counting for the grade. This is the way most MOOC sites do it.

(Of course, if the university servers went down, you should take this into account.)

This earlier question seems to be similar: How to deal with failing a student? (And I gave the same recommendation there.)