Silly Putty™ Armor

BAE Systems^* has developed what it calls a liquid armo(u)r, that is supposed to be proof against bullets:

A liquid armour has been shown to stop bullets in tests carried out by UK scientists at BAE systems in Bristol.

The researchers have combined this "shear-thickening" liquid with Kevlar to create a new bullet-proof material.

The company is keeping the chemical formula of the liquid a secret, but it works by absorbing the force of the bullet strike and responding to it by becoming much thicker and more sticky.

The BAE scientists describe it as "bullet-proof custard".

In theory, this is fairly simple, it's a non-Newtonian fluid, and you can see this behavior with silly putty™ or water and corn starch.

Of course in practice, finding the proper characteristics, and my guess is that a lot of this has to do with the peculiarities of the interaction between the Kevlar fibers and the fluid, is the tough part.

Seeing as how a body part won't be moving at more than about 20 m/s, and a bullet would likely be moving at more than 100 m/s even at the extremes of range, developing a material with the appropriate characteristics is certainly technically possible.

^*Full disclosure, I worked on the Future Recovery and Maintenance Vehicle, FRMV, "wrecker" variant of the FCS-MGV^† from 2003-2006 at United Defense (later BAE Systems after the Carlyle Group sold me to buy Dunkin Donuts).
^†Future Combat Systems-Manned Ground Vehicle. These are the ones that are the tanks and APCs. As opposed to the various unnmanned vehicles, networking technologies, etc. that form the full FCS along with the MGVs.^‡
‡Yes, I have worked everywhere. Maybe I can't hold down a job, but more likely this has been my role as "technical hit man", where you are parachuted in to take care of a specific need.