Is there any convincing evidence for the existence of nanobacteria?
There are two convincing papers (1,2) arguing that the observed "nanobacteria" are in fact mineral/protein complexes and not any living organisms.

In (1) Martel and Young created very similar looking particles from calcium carbonate in vitro. The calcium carbonate precipitations can even look similar to dividing cells:


  After 5 days of serum incubation, a white precipitate formed and
  adhered on the bottom of each flask. When examined by scanning
  electron microscopy (SEM) and transmission electron microscopy (TEM),
  the precipitate was seen to consist of particles with a size ≈500 nm
  (Fig. 1 A, B, and E), an observation consistent with normal bacterial
  morphology, which peaks at one particular size specific to the strain
  on hand. Some forms resembled cells undergoing division (Fig. 1 D,
  arrow).


One strong argument against those nanobacteria being living organisms is that they don't seem to contain any nucleic acids. Of course non-DNA based organisms are theoretically feasible, but that would be an absolutely extraordinary discovery.

The autors of (1) also observed these particles after filtration through a 0.1um filter. They argue that this size could not be sufficient to contain all the cellular machinery bacteria need:


  Because an earlier workshop commissioned by the NAS had suggested that
  the minimal cellular size of life on Earth must exceed 200 nm in
  diameter to harbor the cellular machinery based on DNA replication
  (19), it is unlikely that nanobacteria represent living entities
  unless they contain some other type of replicating mechanism.


(1) Martel J and Young JD, Purported nanobacteria in human blood as calcium carbonate nanoparticles. PNAS 2008, 105(14):5549-54

(2) Raoult D. et al., Nanobacteria Are Mineralo Fetuin Complexes, PLoS Pathog. 2008, 4(2): e41.