Earlier quoted context omitted.
If you ask a rocket scientist to make you a car that passes the crash test ratings alone, they'll build you a tungsten dump truck. If you're going to win, you might as well reach the destination without noticing that you've had a collision.
I believe one of the tests is crashing into a wall that yields very little, so the tungsten dumptruck would fare poorly, exciting as it would be to drive. EDIT: this one: http://youtu.be/V5R80yUUVNk
A single solid tungsten dump truck would have almost twice the mass of the wall. Even with a completely inelastic collision, the wall's going to shatter and move. For a 35 mph collision, I think you're right, that wall would ultimately bring the truck to a halt from friction with the ground. At higher speeds, I think the truck might make it through.
Tungsten is quite dense (19,300 kg/m^3). Such a truck would have a mass of >400 metric tons (2 x 3 x 4 m x 19 tons/m^3), or >247 Toyota Tacomas (2013 extended cab, curb weight 3560 lb --> 1618 kg).
The truck would be terrifying to drive, once you got it going. I don't know how you'd turn.
Alas, Youtube is short on tungsten dumptrucks, but this may suffice. In a demonstration, a truck uses its brakes to stop after obliterating a few cars. The truck is driven by a real person.