An extension in the case of illness is possible. You will have to file an application and turn it in to the examination board and show a proof of illness through a medical certificate. You will need a signature from the examination board and then you will have to send the form to the examination office.
This is the application form: Antrag auf Verlängerung der Bearbeitungszeit mit inkludiertem ärztlichen Attest
It can be sent via post to the examination office or turned in the post box at the office. You will be informed via e-mail about your new deadline.