At 88.4 degrees latitude an average speed container ship making about 25 knots could circumnavigate the globe in 24 hours. This is around 140km from the north pole.

88.4 degrees lattitude

Of course, this area is almost entirely covered with sea ice every year and so would require an ice breaker ship to travel through. But it would allow someone to stay at the same time relative to longitude constantly as they navigated the world and then instantly head into the next day.