Source

This fact is calculated from data recorded in OpenStreetMap. The point that achieves the furthest distance from a post office is here, with the nearest to it being this, this and this.

If you know of a post office that disproves the claim, you can edit OpenStreetMap (here's how) to make sure that item is listed on the map. (NB this website is refreshed occasionally from OpenStreetMap, but not instantly.)