Is it possible to stop duplicating docker images?

We have On-premises software docker image.Also, We have licensing for application security and code duplication.

But to add extra security is it possible to do any of the below ?

1. Can we lock docker image such that no one can copy or save running container and start new docker container in another environment.
2. or is it possible to change something in docker image while build that may prevent user to login inside container.

Goal is to secure docker images as much as possible in terms of duplication of the docker images and stop login inside running container to see the configuration.

No. Docker images are a well known format with an open specification that is essentially a set of tar files and some json metadata. Once someone has this image, they can do with it what they want. This includes running it with any options they'd like, coping it, and extending it with their own changes.