Have we detected galaxies which have red-shifted beyond the visible light range? According to this answer it is possible for galaxies' light to move beyond the visible frequencies due to redshift:

It is possible that eventually the light from them could move into the infrared and even 
the microwave in extreme cases.


Could this have already happened?  Have we already looked for it?  Is there a telescope with the instrumentation to detect galaxies' 'light' at these wavelengths?
Yes, of course. Many, many examples. Telescopes work in the infrared, far-infrared and there are even samples of galaxies that are selected on the basis of their mm emission.

The most distant galaxies detected now have redshifts of 7 or more. This means the wavelength of their light has been stretched by a factor $1+z$ - i.e. by a factor of 8. Thus light in the visible range, say 500nm, now appears at wavelength of 4 microns, in the infrared.

Telescopes that work in this range include the Spitzer space telescope; and many ground-based telescopes. Observations of highly redshifted galaxies are routinely made at infrared wavelengths on telescopses all around the world.

Galaxies are also detected in the far infrared by the Herschel satellite or at mm (getting on for microwave) wavelengths by JCMT or the ALMA telecope.