That means when humans finally colonize Mars, there will be two FaceBooks, FB Mars and FB Earth, with any communication between them running at a lag.

I bet there will be AI research to create bots to act in "real time" on your behalf on the opposite planet. If the artificial agent can imitate your own actions and judgements, it would be a good way to surpass the speed of light. It will be necessary to build these agents because other interested parties could use the time lag to unfairly gain an advantage over you. The more we explore, the larger the time lag. With Jupyter it is 33-53 minutes. With Pluto it's 5h. The nearest star - 4 years. The Moon is at 1.3s.