The term "being" carries the meaning that the entity is biologically alive, and it lives by itself in its context. That is, that it doesn't depend on others to survive.
You will normally find also "living being", but there are no "non-living beings". Those would just be "entities" or things, not "beings".
Of course, such definition holds many assumptions. For example that we anyway need of a functional ecosystem, that the conditions to live are given, and that survival is assessed on the short term: you are a living being because you don't need of other humans to survive in the short term, now, this day, this week. Because on the long term, we all depend on the others.
Therefore an arm is not a "being", it is not an entity that can live one day without being attached to a living being.