Could an asteroid land slowly on Earth's surface?
If the asteroid is in parallel to the orbit of the earth and at rest it will feel the gravitational attraction and will fall with velocity growing as $g\cdot t^2.$ This force will be there whatever the angle and velocity of the asteroid, centrifugal forces may make it miss the earth in a parabolic orbit, or be caught in an elliptical as the path of the satellites. To avoid falling on the earth with great velocity it would need not only to have a small velocity relative to earth but also an acceleration equal or larger and opposite to the acceleration of gravity.
Well, technically, the answer is no as the other answers and comments also say.
The approach speed can not be less than escape velocity. But in order for such a thing to happen, nature has to be really creative and totally in our favor. For example, the asteroid can have a very very lucky combination of these:
The asteroid has right kind and amount of fluid in it that starts jetting out steam at just the right times and right angles.
The asteroid is parachute shaped with appropriate strength and falls at an appropriate angle.
Again, it would be a miracle, so, please do not hit me.
As we may be lucky due to a three body interaction with moon, this is taking the luck to kind of extreme.