Yes you can!
If you tap on any of the icons after playing an Event, a permanent private link will be generated:
For iOS users, please tap on the icon on the upper right:
You will see all the options:
The Event will also be saved to Moment:
You can also open the link with the Chrome app, long press the screen, and download the Event to your device.
Alternatively, you can save the Event to your PC by following the instructions in this post!