I would like to run integration test while I'm building docker image. Those tests need to instantiate docker containers.
Is there a possibility to access docker inside such multi stage docker build?
No, you can't do this.
You need access to your host's Docker socket somehow. In a standalone docker run command you'd do something like docker run -v /var/run/docker.sock:/var/run/docker.sock, but there's no way to pass that option (or any other volume mount) into docker build.
For running unit-type tests (that don't have external dependencies) I'd just run them in your development or core CI build environment, outside of Docker, and run run docker build until they pass. For integration-type tests (that do) you need to set up those dependencies, maybe with a Docker Compose file, which again will be easier to do outside of Docker. This also avoids needing to build your test code and its additional dependencies into your image.
(Technically there are two ways around this. The easier of the two is the massive security disaster that is opening up a TCP-based Docker socket; then your Dockerfile could connect to that ["remote"] Docker daemon and launch containers, stop them, kill itself off, impersonate the host for inbound SSH connections, launch a bitcoin miner that lives beyond the container build, etc...actually it allows any process on the host to do any of these things. The much harder, as @RaynalGobel suggests in a comment, is to try to launch a separate Docker daemon inside the container; the DinD image link there points out that it requires a --privileged container, which again you can't have at build time.)