Yes, you can export and save any work you've done on the whiteboard. Just click File, then select Export Board as Image to save individual whiteboards during the lesson.
Tutors and the students will also be presented with an option to save all of the whiteboards from an online lesson once the tutor clicks the "End Lesson" button.
Be sure to also save your work from the Text and Code Editors separately! These will not automatically save when you download your whiteboards.