The country in question was New Zealand, with the Doctor's very first appearance outside the United Kingdom to be broadcast by Christchurch's CHTV-3. The episode was shown at 7:57pm, sandwiched between news programme NZBC Reports... and a documentary about Dr. Gordon Seagrave, The Burma Surgeon Today, and was introduced by the weekly magazine The New Zealand Listener as:
The first of a new adventure series about an exile from another world and a distant future, travelling with his granddaughter and two London school teachers through time and space. Starring William Hartnell as Doctor Who and Carol Ann Ford (sic) as his granddaughter. In tonight's episode Ian Chesterton and Barbara Wright, two school teachers, decide to try and find out more about one of their pupils who is puzzling them.
You can read the country's introduction to the Doctor from the Listener below.