Absolutely. I power down, and take somewhere without internet or WiFi connectivity, and local record works fine. For example, take a look at this thread:
Other than the clock being wrong because the camera could not reach a time server, it recorded fine. The video shown was recorded onto the uSD card. I have done similar many times. Always worked. The V2 I have in my pickup truck as a dashcam is out of WiFi coverage as soon I am more than a few houses from home. Although it is normally powered 24x7, it does at times get unplugged (usually because I needed the 12 volt outlet for something else). Always comes back up fine.