Yes, whatever you stored inside docker container is permanent. But one thing you should understand, difference between ephemeral and persistent storage. Let's discuss with an example.
Say, You have a webserver inside docker container. Everyday lots of data is uploaded in your server. Now if you shut down your container, your data will not erase from your container. You can start your container and check it.
But, after some days if you don't require that webserver than you will terminate that container and now your data is permanently gone. So, these types of storage is called ephemeral storage, that depends on OS.
But, if you terminate your container and this time you require that database, for that you have to use persistent storage. This types of storage will not depend on OS.
Hope these will clear your doubt regarding storage in docker.