Translink Metro route 93 in Belfast, Northern Ireland, may fit the description.
It runs once daily Monday to Friday in each direction, timed for a 9am-5pm working day, providing a service from the City Centre to an office park on Newforge Lane, which is one mile from the nearest other bus stop (for the 8a/b/c service) on Malone Road.
It is officially a public bus, but it always runs with the destination screen either blank or displaying "Private". The entire route is pick-up only on the outward journey. There is no matching set-down only restriction on the return journey, and the route appears on the stop flags and real time display like any other bus to the City Centre - but it will never pick up passengers.
Also, for some bizarre reason, on the outward journey, instead of picking up opposite the city's main bus and railway station like every other bus to South Belfast does, it is instead routed via the back streets behind City Hall and picks up at the Ulster Hall (a concert venue) around 250 yards away from the station. But like every other bus, it does serve the station on the return journey. I can't see any logical reason for this.
Apart from the slight detour on the outward route mentioned above, it follows the 8b/c route between the City Centre and the junction of Malone Road, Balmoral Avenue and Newforge Lane. It then runs the full length of Newforge Lane, whereas the other routes continue on Upper Malone Road then Finaghy Road to Erinvale Estate (8a/b) or Ladybrook Estate (8c)