Do bacteria die of old age?
This is a interesting question and for a long time it was thought that they do not age. In the meantime there are some new papers which say that bacteria do indeed age.

Aging can be defined as the accumulation of non-genetic damages (for example oxidative damage to proteins) over time. If too much of these damages are accumulated, the cell will eventually die.

For bacteria there seems to be an interesting way around this. The second paper cited below found that bacteria do not divide symmetrically into two daughter cells, but seem to split into one cell which receives more damage and one which receives less. The latter one can be called rejuvenated and seems to make sure that the bacteria can seemingly divide forever. Using this strategy limits the non-genetic damage to relatively few cells (if you consider the doubling mechanism) which could eventually die to save the others.

Have a look at the following publications which go into detail (the first is a summary of the second but worth reading):


Do bacteria age? Biologists discover the answer follows simple
economics
Temporal Dynamics of Bacterial Aging and Rejuvenation
Aging and death in an organism that reproduces by morphologically
symmetric division.