Is it possible to start a new game in Metal Gear Solid V: TPP?

No, you cannot fully reset your progress in this game. Developer Konami has not implemented a feature that allows you to start over. Other options like deleting your saved data off your hard drive also do not reset the game as your progress is restored when you boot the game and sync with Konami servers.

For a partial success you can disconnect your console or computer from the internet, delete your save off your hard drive, and loading the game in the disconnected state to prevent progress from syncing with the online servers. This will generate a new game on your local device, and when you reconnect to the internet the save stored on Konami servers seems to be overwritten.

There's a big caveat though: your online progress is always preserved. This means FOB development, online GMP, vehicles, gun emplacements, and resources all remain as they are part of your online progress. If you don't wish to replay with access to the 43 Magloader tanks you acquired throughout your previous playthrough then you must remain disconnected to the internet and never sync with the online servers. Nevertheless, your research/development, staff size, and mission progress will be reset if you delete your save while offline - you'll have a surplus of supplies and not enough staff to handle them.

You may want to contact Konami support and see if they can clear your online progress, but they won't be able to. I opened a support ticket with Konami to reset my online progress and this is their response:

Hello, Thank you for contact KONAMI Customer Support.

In order to have a fairly managed service, we are not able to provide support regarding rewinding online data to the initial state. We apologize that we were not able to meet your expectations regarding this matter.

Thank you for playing METAL GEAR SOLID V: THE PHANTOM PAIN.

No dice. So what can you do about this? Well, not much. You'll have to either start a new game with a overwhelming amount of supplies from your previous playthrough or find some way to get rid of them before deleting your local save. I've found the fastest method to be to sell my vehicles and gun emplacements then visit a remote corner of Afghanistan and call a nonstop series of GMP-consuming air bombardments. That still takes a significant amount of time to reduce your GMP to zero after selling a nation's worth of military surplus and I think it wouldn't be worth it for most people.

If you really want to restart your progress in MGS V:TPP you'll have to settle for resetting mission and Mother Base progress, because you can't reset everything.