Death, in the biblical sense of the word, is always a spiritual separation. Physical death happens when one's spirit separates from the body. Spiritual death is when the spirit of a man (human) is separated from God. It is the later of these two that definitely would not have occurred prior to the Fall. Whether physical death was possible is an open question. The Tree of Life existed in the Garden of Eden and so anyone who ate of that Tree certainly would not have died physically but those who had not, it seems, would have been able to die physically. Otherwise, the Tree of Life would have served no purpose.
Plants, of course, do not have a spirit at all and so could not have died in the sense spoken of above and the extent to which the above applies to animals is speculation. Certainly some form of decay would have occurred. The banana peels that Adam and Eve would have discarded would have been consumed by microbes and recycled into the Earth, perhaps in a process identical to what happens today. And things like insects, worms and the like would have likely been on the menu for things like birds and rodents, and birds and rodents would have been on the menu for larger animals, etc.
