The technology will initially be on trial at Rievaulx Abbey in North Yorkshire Moors National Park, Wasdale Head in the Lake District and two sites in Snowdonia National Park, Wales, said the Department for Digital, Culture, Media and Sport (DCMS).
The test sites will use equipment supplied by Starlink, a part of Mr Musk’s SpaceX company.
It uses a network of low-Earth-orbit satellites to provide internet signal in places where there is limited ground infrastructure, or where it would be expensive and difficult to put it in place.
The department said recent tests showed that in many locations, Starlink satellites can deliver internet speeds of up to 200 megabits a second – four times faster than the UK average broadband speed of just over 50Mbps.
The government said it was continuing to look at the ability of the system, and other solutions and services with different suppliers.
Mr Musk has become a polarising figure in recent months after his takeover of Twitter, which has divided opinion because of his push for less content moderation and more free speech on the social media platform.
Elon Musk - in pictures
“High-speed broadband beamed to earth from space could be the answer to the connectivity issues suffered by people in premises stuck in the digital slow lane," UK Culture Secretary Michelle Donelan said.
“Ensuring everyone can get a quality internet connection is crucial to our levelling-up plans and these trials aim to find a solution to the prohibitively high cost of rolling out cables to far-flung locations.”
The government announced announced a contract worth more than £100 million ($121 million) to build gigabit-capable broadband connections for up to 60,000 rural homes and businesses in Cumbria.
Northern Ireland provider Fibrus will carry out the work, which is the biggest contract so far under the government’s £5 billion Project Gigabit scheme to take faster internet to all parts of the UK.
Elon Musk's 2022 - in photos
It was also announced that the value of vouchers available under the Government’s Gigabit Broadband Voucher Scheme will be tripled next year.
That means eligible homes and businesses in rural areas will be able to apply for up to £4,500 to cover the costs of installing a gigabit-capable connection.
The scheme is open to homes and businesses in rural areas where: existing broadband speeds are less than 100mbp; a commercial network is unlikely to be built there in the near future; or there is no government-funded contract planned or in place to improve connectivity.