The first car cannot avoid the deer, true, but this condition will not cause a pileup.
Think about the physics. The inelastic collision between the deer and the car will marginally slow the front car down, true, but only slightly. (since a car weighs 2000 kgs and a deer weighs less than 100, for an estimate). So the combined car-deer vehicle will be going only slightly slower.
Ok, so now the car that is about to hit the deer applies maximum braking force. It begins to decelerate at a rate limited by friction between ground and car. This friction is independent of the mass of the car, for reasons I can't fit into here.
The moment it hits the brakes, the car behind it will see the distance between the two begin to decrease. They are "bumper to bumper", or within 1 meter of each other. The car behind will apply maximum braking force the very moment a single cycle of it's control loop happens (probably 1/1000 or a second or so).
The car behind that will do the same, and so on.
As long as no car in the pack has significantly better brakes than the other cars, no one will hit anyone. Even if a particular car does have better brakes, the collision will only do slight damage, as the relative velocities will be low.
Contrast this to what can happen in a real highway, where a car in front can have time to decelerate to a stop in some cases, and the cars behind may be driven by a distracted driver who does not see the stopped vehicle in time. The collision happens at highway speeds between the trailing car and the stopped car. This, in some cases, will be fatal.