Can 3D printers print details in the 1/10 of the micrometer for metals?
You can get the 0.1 micron (100 nm) resolution with a 2-photon 3D printer, but only in a polymer resin.  Nanoscribe, in Germany, pioneered this technology, see Mechanical Microstructures.  

Their commercial printer, the Photonic Professional GT, is about $350,000 US with software and accessories. There is some work being done to replicate the 3D printed polymer in metal using electroless plating or ALD (atomic layer deposition).  Other techniques are in development.

None of the direct metal 3D printing processes come close to your 0.1 micron resolution, although the field is rapidly evolving.  Only a couple of years ago, direct metal 3D printing was all based on powder bed fusion.  Now binder jet technology as been adapted to metals and, very recently, Xjet has developed a nanoparticle 3D printer. It prints "ink" composed of metal or ceramic nanoparticles suspended in a liquid.  The minimum layer thickness is 1 to 2 microns.  They have not released XY resolution data yet.

A good overall reference for the various 3D printing techniques (including Xjet, but not Nanoscribe) can be found at Explaining The Future - 3D Printing.