Here is some way to achieve this.

The idea is to save images that you want to use, and then to import this images during the container startup process.

For example, you can use a folder images where to store your images.

And you have to use a customized entrypoint script that will import the images.

So the Dockerfile will be :

FROM docker:19.03-dind

RUN apk add --update --no-cache bash tini

COPY ./images /images
COPY ./entrypoint.sh /entrypoint.sh

ENTRYPOINT ["tini", "--", "/entrypoint.sh"]

(tini is necessary to load images in background)

And the entrypoint.sh script :

#!/bin/bash

# Turn on bash's job control
set -m

# Start docker service in background
/usr/local/bin/dockerd-entrypoint.sh &

# Wait that the docker service is up
while ! docker info; do
  echo "Waiting docker..."
  sleep 3
done

# Import pre-installed images
for file in /images/*.tar; do
  docker load <$file
done

# Bring docker service back to foreground
fg %1

FYI, an example on how to save an image :

docker pull instrumentisto/nmap
docker save instrumentisto/nmap >images/nmap.tar