The reason that you get slip at even the smallest forces results not from the fact that the tire is slipping against the ground, but that the tire is elastic. There is no way to completely eliminate slip with an elastic tire. Let's see why this is.
To measure the slip, lets put twenty little green splotches of die evenly spaced on the circumference of the tire, as shown in the following figure:
Now if this tire is rolled along the ground with no tangential force between the tire and the ground, we expect the pattern left on the ground to look like this:
where the spacing between the marks on the ground is the same as the spacing of the marks along the circumference of the tire. Now we know that since there are twenty splotches along the circumference of the tire, the distance separating twenty (well twenty-one, technically) splotches on the ground corresponds to the distance the car travelled in one revolution of the tire. Since the spacing of the splotches on the ground is equal to the spacing along the circumference of the tire, we see that distance the car travels in one revolution of the tire is indeed one circumference of the wheel. Thus there is no slipping when there is not tangential force between the tires and the ground.
Now imagine there is a tangential force between the tire and the ground (as will inevitably happen due to air resistance). What happens to the tires? Well the road is providing a force on the tire. This will cause the tire to deform and the bottom of the tire you will get scrunched up in the direction of the force. This is shown (exaggerated) in the figure below:
As the wheel rolls forward (to the right), the scrunched up parts will leave their mark on the ground. This means that our little green splotches in this case are twice as close together, as shown below:
When the tire makes a full revolution, it must leave twenty splotches on the ground. On the other hand, these splotches have only half the separation that they did before. Thus the car only moves forward by half of a tire circumference when the wheels rotate once. Therefore, we conclude that the wheel has slipped.
Since there will always be some scrunching given a non-zero tangent force, you will always get slip for a nonzero tangent force.
Now for high enough slip ratios, the wheel will actually slide across the pavement, but until you get to the this point, static friction is still in play, so the car is accelerating from static friction.