If I understand the question correctly, the following two options are applicable in my experience:
1) Silver nanoparticles of 10nm or larger are centrifugally pelleted at 12-15,000rpm over 1 h-90mins, leaving silver nitrate in solution. Wash the pellet several times. For larger silver nanoparticles, try a slower speed. Ideally a soft pellet will be easier to resuspend in solution (water?).
2) Size exclusion chromatography of capped (coated) silver nanoparticles (such as using Sephadex G25 chromatography media) will easily purify the nanoparticles first, before washing off salts from the media.
if particles are dissolved you will not be able to separate them (solution of ions). For nanoparticle suspension do the calculation based on Stokes law. Make sure that during washing particles do not get destabilized.
I used 17000 g for centrifuging and washing the solution, but the supernatant was still yellow. Is that mean that certain silver nanoparticles was still in the supernatant?