How can I uninstall the BBC Player app?

From this page from the Netflix FAQ:

  1. Begin from the PS4 Home Screen. ...
  2. Navigate to the TV & Video section and highlight Netflix.
  3. Press the Options button on the PlayStation controller.
  4. Select Delete.
  5. Select OK.

Replace "Netflix" with "BBC Player" and you're good to go.

Alternatively, you could go to the Settings menu and go to System Storage, then find BBC Player in your list of apps and then use Options > Delete as above.