No, not all eukaryotic cells have only one nucleus. While most eukaryotic cells, such as those in animals and plants, typically contain a single nucleus, there are exceptions.
For instance, some eukaryotic cells are multinucleate, meaning they have multiple nuclei. A common example of this is skeletal muscle cells in animals, which are long and can contain many nuclei to support their larger size and greater demand for protein synthesis. Another example is certain fungi, like those in the genus Rhizopus, which also exhibit multinucleate structures during their growth.
Additionally, some cells can lose their nuclei altogether. For example, red blood cells in mammals do not have a nucleus when mature, which allows for more space to carry oxygen.
Therefore, while one nucleus is the norm for many eukaryotic cells, it is important to recognize that nature has diverse ways of organizing cells, and there are notable exceptions to this rule.