No. There is no such capability in lightweight containers or in common operating systems like linux or windows.
Let me ask…can you think of such a capability offered on a non-docker system? For example, generally, in windows or linux, can I take a snapshot of a process, reboot the system, and then restore the snapshot to continue from there? I’m sure there is no such thing for processes running on a host. Do you know of one?
Most applications are designed to store their important state data in a persistent manner so the data will survive a crash or shutdown. Then, after reboot, as a new instance of process is started, it reads the important state data from storage. Using the state data, the application figures out for itself where it left off.
I could be wrong in understanding what you are seeking.