The A962 is allocated to part of West Castle Street (the eastern end of which is also called just Castle Street) in Kirkwall, Orkney. It's about 100 metres long and links the A960 with the A963. It's too short to be numbered on the current Landranger map, but the number was shown on the 1946 OS 10-mile Road Map (although the road was too short to display)!
With the semi-pedestrianisation of Albert Street, technically the A960, and certainly with a one-way system operating on it, traffic is now signed to use the A962 in place of the A960 when passing through the centre of Kirkwall. This is despite traffic from the pedestrianised section of Albert Street getting priority over A962 traffic!