No, it is not possible that a student submitted work that was subsequently lost.
A student's answer is saved each time they click Check Answer within an assignment. Or, if the assignment is in Test mode (which disables the Check Answer button) the student's answer is saved each time they navigate from one question to another. The timestamp of the last submitted answer is recorded and shown at the top of the assignment. As an instructor, you can see that timestamp, as well as every response the student submitted in the form of a Correct or Incorrect response tab. If there is no timestamp or response tab, then the student did not submit any answers (or the assignment was purposely reset by an instructor in the course). For more information, see the help article Viewing and Resetting Work.
In many cases, you can check the access logs to prove that the student never even accessed the assignment prior to its due date. However, the presence of a log entry from before the due date does not prove that the student completed the assignment, only that they viewed it. The response tabs (labeled Correct or Incorrect) within the student's assignment are the proof of their work. Note that the Solution tab is not a response tab. The Solution tab is always visible to the instructor and does not represent an answer submitted by the student.
If the student truly believes that they completed the assignment, they must have been logged in as someone else at the time (maybe in computer lab and they forgot to log the previous user out). If you'd like to give the student the benefit of the doubt, we suggest granting them an extension and warning them to be careful about logging out/in when using a shared computer.