Reykjavikurflugvöllur (RKV/BIRK), located at Þorragata 10, features twice in the television production. Firstly showing TF-FIM taking off when Stewart and Graham fly to Húsavík, secondly when Jack Case arrives in TF-EPP where Slade greets him. TF-EPP is seen taxiing in the area opposite the Loftleiðir hotel; near what is now the BIRK flight services office.
Isavia own Reykjavikurflugvöllur with BIRK flight services being the ground handler. The first flight from the airport was on 3rd September 1919, with the take off of an Avro 504, the first aeroplane in Iceland.
An article in the Icelandic newspaper Morgunblaðið on 4th September 1919 describes the event:
First flight in Iceland
Capt. Faber flew twice yesterday – Flugfélagið announced yesterday morning that the aircraft would be on display at 8 o’clock last night. No promises were made that it would fly. The engine had never been tried and though nothing had proven wrong with it when it was put together there was a possibility that there might be unforeseen problems and it wouldn’t be able to fly. We expected that the flight would not take place until the end of the week.
Innovative spectacle – But at 5 o’clock yesterday an unexpected event occurred at the south end of the airport. An immediate and unexpected decision was made that the aircraft would fly and without anyone knowing, Faber got into the cockpit, sat at the console and took control. The aircraft ran a few dozen fathoms down the meadow and then she finally rose from the ground like a swan flying up from the water. The sound of the motor could be heard in the town and the people began to look around. And all that heard, looked up and saw the machine fly through the air like an enormous bird; stable as a path of light and turning sharply, after a while it landed on the grass.
Another flight – At 8 o’clock last night a lot of people had gathered to see the machine, and there were many who had never seen an airplane before, some waiting more than 50 years to see this strange site. Members of Parliament and town council officials had been invited there and at 8 o’clock the aircraft was driven from the hanger towards the crowd. After a while the airline chairman Garður Gíslason came to the plane and gave a speech to those who were gathered. This was a special occasion; we have the most up to date transport in the world. It was here, a small nation in a large country, a nation that has endured to acquire the main transport environment, rail. The aircraft would now be the preference of transport, not bridges or roads, it would equal the mountains valleys, go over lava fields and glaciers. He mentioned the persistence the pioneers of flight had shown, and that the new management had always been ready to strive until that perfection was achieved, the modernity of flight. He asked Iceland to provide the example of progression that would be necessary for it to be successful and become a significant part of the country’s transportation system. He named the Icelandic Parliament as well as individual men whose dedication and intercession finally made Icelandic aviation possible and wished for its continued success.
Following this, Faber took the helm and aircraft mechanics began to turn the propeller. The motor soon came to life and had only been running a few moments before the aircraft started moving, slowly at first and then cantering across the field.
Up! – People looked expectantly at the machine, and when she left the ground the air resounded with applause and the shouts of many. A number of people had never seen an airplane take off before, and it had quite an effect on some people. Not one voice. Horses next to the airfield staring amazed at this marvel. And one dog was excited.
The aircraft rose a little in the air reaching about 500 meters and continued. Occasionally the pilot throttled back the motor, casting the nose of the machine down a few meters and then leveled again. After flying a while, the aircraft reduced height again and flew a few meters above the heads of the people in a straight line right across the field. She went up again and flew for a while longer before landing. In total, it was in the air for a quarter of an hour. 
The article may be viewed here.
Images © BBC Scotland, © The Bagley Brief.
1. Morgunblaðið (1919). ‘Fyrsta flugið á Íslandi’ (4th September 1919 p. 1).