This actually depends on the procedure your company has lead down in the MEL and which is required to be approved by the respective DGAC / CAA.
Inside JAR it is possible that you have the option to have the Tech. Director to approve an extension for the very same period as the MEL originally limited the item; i.e. in your case, the Technical Director may approve to operate with the deferred rectification until 5th January 23:59
Other operators (due to authorities) need prior approval from the respective CAA to defer the rectification for the same period or shorter, as case may be decided.
In both scenarios the extension has to be mentioned and signed by an authorized engineer near the entry in the ADD listing.
In any case, the aircraft is not allowed to be dispatched nor operated outside the approved and signed deferred rectification period.