Not now. The smallest microbes are the ultramicrobacteria, they are only 200 nanometers in diameter, but that's not really small enough to call them "self replicating nanobots".
But - in the past life must have had much simpler precursors. There is no way that anything as complex as modern life could have arisen in one go from simple chemicals.
One workshop on this came to the conclusion that the earliest precursors of life were probably about 40 nanometers in diameter. Page on nasa.gov
This would be large enough to contain all the machinery of replication of a small microbe with only five genes (say). It might have RNA only without DNA, so doesn't have to have the complex machinery needed to translate one to the other - and could use "Ribozymes" made up entirely of RNA fragments for the enzymes to catalyse critical cell reactions instead of the much larger ribosomes, which are made up of RNA combined with a protein, by way of example.
And there is also a lot of work on the "synthetic minimal cell" - some of those consist of only a few hundred atoms.
So, it seems reasonably likely that Earth did have nanoscale life in the past. Other planets in the solar system may well still have it. Also possible that our modern Earth life is unnecessarily cumbersome, that it followed a particular line of development (like the marsupials in Australia) and that elsewhere life is simpler and more streamlined.
Some people think that Earth may still have nanoscale life. It would be very hard to detect if it does exist, if it exists as a "shadow biosphere" that only interacts very weakly with normal life. Especially if it is not based on DNA and maybe not even based on RNA but some other informational polymer such as PNA or TNA.