If you are from the UK, you should probably clarify what you mean by a "Public school" since a reader from the US probably interprets that differently than what you mean.
Yup. A huge number of rich foreigners send their children to UK fee paying schools. Lots of Middle-Eastern, Russian and Chinese. Something like 15% of current world leaders were educated in the UK.
Certainly caused some mumblings about the general character of some of the most prestigious institutions like Eton. It would be easy to put that simply down to xenophobia but there is a relative difference between the established culture and the new. British aristocratic elites are generally (and of course this is all relative) far poorer than the children of Oligarchs and party officials. Generally the attitude of the aristocratic Brits is to keep their head down and not be flashy, whereas some of their classmates literally have a private jet for their use on weekends.
Its what I have been told happens in my home towns Public schools - one of Barack Obamas roomies at Harvard came from one of them.
As suggested a in the UK Public School is what an American would call a private one ie one where rich people send there kids