Create and publish a Docker image

⚗️ Make sure user can write to cache #189

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Run time

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Job	Run time
build-and-push-image (3.14)	4m 25s
build-and-push-image (3.10)	4m 19s
build-and-push-image (3.13)	4m 26s
build-and-push-image (3.12)	4m 14s
build-and-push-image (3.11)	4m 20s
	21m 44s