Not a mythril shirt, but something just as nifty.
Researchers at the University of Illinois at Urbana-Champaign have figured out how to use microphotolithography techniques to make the world's smallest chainmail. The links are made of vapor deposited copper, and each link is 500 microns in diameter. Testing shows that the micro-chainmail has the tensile strength of nylon, can stretch 1/3 of its resting length, and is extremely flexible. As flexible as 'real' fabric, in fact.
Oh, and did I mention that it conducts electricity normally?