Do some eukaryotes reproduce asexually

Actually yes some of them do. Not all of them do though, most of them produce sexually.

Yes, some eukaryotes, such as protists and unicellular fungi, reproduce asexually through binary fission. Binary fission is the process of a cell creating two copies of its own DNA and then pinching in half, resulting in two identical "daughter cells." The parent cell is gone after the process, having been split into two new cells (albeit with the same DNA).