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Ford did not build any Ford Anglia 105E Convertibles themselves, but there is a rumour that the Convertible specialist company "Crayford" may have built either a prototype or a small run of production cars.

The "legend" of a "Crayford" convertible is first mentioned in the May 1982 edition of "Practical Classics" magazine, which shows the above photograph with the caption "An Anglia Convertible thought to be one of about 20 made by Crayford"

Micheal Allen's book "Anglia, Prefect, Popular" of 1986, has a similar photograph, but this time it is captioned "a convertible Anglia, thought by its present owner to be a one-off by Cayford"

I have not been able to verify this, as no mention seems to be made of an Anglia in any of the Crayford literature I have seen.

In 1962, the Italian stylist Pietro Frua produced a design for a Ford Anglia Spyder (along similar lines to his other Spyder designs for Opels, Fiats etc.) A prototype was built in Italy which appeared in the "auto italiana" magazine in 1963. "Official" photographs were taken of the car and were shown to Ford of Italy. For some reason this stylish car never made it to production. (Possibly because of the slow sales of the Ford Anglia Torino)

The lack of an official Ford Anglia 105E Convertible has not stopped enthusiasts building there own over the years

If Crayford have indeed built any Ford Anglia 105E Convertibles it would make them a very rare and unique variant of the Anglia family.
The total number of 105E Convertible's built is unknown