"yes, anything which grows has dna"

There are viruses who have their genetic information written soley in RNA, a related nucleic acid.

Prions are infectious agents composed soley of protein. (think made cow disease).

Whether or not to consider these infectious agents living or non-living is another matter.....but they certainly have the capacity to grow in numbers.

Crystals can also grow.....

So I'd say the best answer is simply "yes it does"