Building a mass-market electric vehicle and colonizing Mars aren’t ambitious enough for Elon Musk. The billionaire entrepreneur now wants to merge computers with human brains to help people keep up with machines.

The founder and chief executive of Tesla Inc. and Space Exploration Technologies Corp. has launched another company called Neuralink Corp., according to people familiar with the matter. Neuralink is pursuing what Mr. Musk calls “neural lace” technology, implanting tiny brain electrodes that may one day upload and...