When searching the usual databases, I stumbled over other families that bear the same surname but do not come out of Germany. I would be happy to know more about these from researchers who specialize on regions:
1. Iceland: Grimur Gislason Eyman, b. 1860, d. 1937, is the father of an Eyman family from Selkirk, Manitoba, Canada. I have not so much information about where this family comes from. His middle name Gisla-son is in nordic name-giving tradition "the son of Gisli". Grimur's father is a Gisli Sigurdsson (b.1828); no "Eyman" in his family name and I have no idea where this originally comes from.
2. Estonia/Finland: there are several people with a family name of "Eiman", e.g. descendants from Ludvi Eiman b. 1823. I am still collecting and structuring the data given here.
3. Austria: some Eymann records can be found in Lower Austria/Niederösterreich, e.g. Michael Eymann (1759-1842), Magdalena Eyman (b. 1799), or Wenzel Filiz Eimann (b.1848, m. 1879 with Franziska Fischer).
4. Bretagne region of France: starting with Yves Eyman (b. ca. 1794) m. Anne Kouredand, there are some entries with Eyman names that are not well connected. There is some possibility that the correct name is "Tymen" and that all the Eyman entries from Finistere are transcription errors.
5. I found some entries with the "Iman" surname in families coming from Asia, and even some Jewish entries.
6. According to the Dictionary of American Family Names 2nd edition, 2022, Iman is also a surname derived from a personal name based on Arabic īmān ‘belief faith’ (see the Koran 25).
This is the phonetic name all over the world, and there is probably no connection between any of these. I still even have some isolated families within the German-speaking countries, which I will take up in another post.
