The Martin had originally been purchased by the Mealing Flying Club and had been intended for charter work. Lady Caroline Langdon used some of the proceeds from the sale of a ruby to purchase the aircraft and loaned it to Browning for him to fly to South West Africa. Browning had asked the club to fit an extra tank and told Bunny Hale, the club secretary, that he intended to try to beat the light aircraft record to Cape Town.
There is not aircraft type named the Martin and so it must be considered a fictional type, or a type inspired by a real aircraft but whose name had been changed. One possible inspiration would be the Percival Q.6 Petrel. It was a twin engine six seater and had a speed almost equal to the Percival Gull racing aircraft and so it would have been a decent contender for an air record. Note also the name Petrel, which is a bird, like the Martin. The Petrel was also made of wood. It the later chapters, the Martin wreckage was all burnt out, except for the engines.
The only part of the description working against the Petrel is where Bunny Hale states that the club bought the Martin brand-new. Manufacture of the Petrel stopped with the outbreak of World War 2. The only Petrels available for sale during the time of the story would have to be second-hand ones. Then again, according to Bunny Hale, Lady Caroline/Browning paid two thousand five hundred pounds for the aircraft--that would be a little too cheap a price for something brand new but quite plausible for something second hand from the 1930s.