Yes! Screencastify does not connect to the internet while recording, so you can use it offline.
If Screencastify is connected to your Google Drive, your videos will not save to your Drive until you go back online. They will be temporarily saved locally.
When you finish a recording while offline, you'll see this message in the right sidebar of the Video Page:
I am connected to the Internet, but still see this message
If you are connected to the internet, but you're seeing this pop-up:
please sign out and sign back into the extension to clear this message: