No, you cannot persist the UIAutomation element from the automation client. You will need to store sufficient information about the element (the AutomationID and context are the best, if available) to find it again the next time.
The element is a reference to the object in (probably) some other process, and it can only live so long as that other process lets it. Elements are likely to go away when their windows close, the app closes, etc. Elements such as menus and flyouts are likely to be very short lived, and may no longer exist by the time the client receives a notification.