Can animal cells be broken down further into a living unit?

No, and neither can plant cells, bacterial cells, or most fungal cells.

The giant multinucleated slime mold cells (a type of fungal cell that can be more than a foot in diameter) might be considered a possible exception to this as breaking it down almost always results in making many smaller functional living multinucleated cells. As other cells are not multinucleated already breaking them down precludes their chance to duplicate their genetic material that they need to live.