Hi! Bitbucket uses git under the hood and git is an excellent place to store text files, such as Dockerfile. It does support storing binary files but it is not a good match for Docker images.
You can probably save the Docker image using docker save and commit that - but instead I suggest you take a look for a real container registry. Here are a few (in no particular order):
Looks like Jfrog may be one. They have several articles on Artifactory and Docker integration:
I hope this helps.
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