Did Ashkenazi (or Yiddish-speaking) Karaites exist? Theoretically, such communities could exist in Ukraine and Lithuania. Are there any examples of the same communities?
There were Kariates living in Russia among the Ashkenazi Jewish community but they did not consider themselves any more Jewish than the Samaritans consider themselves to be Jews.
They were known as the Karaylar Karaites. Their presence in what became czarist Russia can be traced back to the 1200s. Unlike Karaites living in countries like Egypt, although they also recognized the divine origin of Torah , and they keep some of the Yomim Tovim etc. , they do not claim to be Jews. Another difference between them and other Karaite groups is that probably are not of actual Jewish descent. They seem to be descended from the Turkic or Tatar peoples of eastern Europe.
In the 1800s they successfully appealed to the czar to to be recognized as distinct from the Jews. Even the Nazis YM"S for the most part did not kill them because they weren't considered Jews.