Is it possible to have undamaged healthy cell, that is dead?
There is a great bit on the podcast "The Infinite Monkey Cage" wherein Brian Cox recurrently asks "When is a strawberry dead?" 

If you think about it, it is a difficult question: when a berry is growing on a plant, it is easy to say it is living because the cells are actively dividing and functioning. But when the berry reaches maturity, the autosomal cells no longer divide. This is called Programmed Cell Death or apoptosis. This is a physiological process. However, one could still argue the strawberry is alive in the sense that the seeds are still viable and can actively produce new strawberry plants. These cells are quiescent. What though, if a strawberry is crushed to smithereens or a disease infects all of the cells? In this case the "death" of the cell has ceased function as in apoptosis, but this is externally mediated. This type of non-programmed cell dysfunction is called necrosis.

So, in direct answer to your question, you must taken into account the mechanism (internal or external) for cessation of function and the permanency. Internal and permanent = apoptosis. Internal and temporary = quiescent. External and permanent = necrosis. External and temporary = ...maybe injury?

To be clear, I've used a multicellular plant as an example (because it is easier to relate to than yeast), but the principles are the same for unicellular organisms.

@Flo made the good suggestion to add references for those interested:

Buttner et al. (2006) Why yeast cells can undergo apoptosis: death in times of peace, love, and war. Journal of Cell Biology

Gordeeva et al. (2004) Apoptpsis in unicellular organisms: mechanisms and evolution. Biochemistry