No sovereign country has its territory entirely or even mostly a mile (1.6 km / 5,280 ft) above sea level.
Lesotho (enclaved in South Africa) holds the distinction of the highest lowest point of any country: 1,400 m (4,593 ft) at the junction of the Orange and Makhaleng Rivers. Its entire territory is above 1,400 m—earning it the nickname "Kingdom in the Sky." But that's still ~2,700 ft short of a mile.
Mount Everest (Sagarmatha/Chomolungma) on the Nepal-China border: 8,848.86 m (29,031.7 ft) = 5.5 miles above sea level. The prompt's claim that Everest is "only about a quarter of a mile" is incorrect—it's over five miles high.
In summary: No country averages a mile high. Lesotho comes closest for minimum elevation. Everest is 5.5 miles high, not a quarter-mile.