You are right; the chalk will continue on an orbit that is consistent with its initial position and velocity when it is released. But it will not move in a spiral. The gravitational attraction of the spacecraft can be ignored.
To begin, take note that objects in higher circular orbits move more slowly than objects in lower circular orbits.
Now assume, for example, that the spacecraft is cigar-shaped, and keeps one end (the "low end" always pointed toward the center of the earth (the way the Moon always keeps one face toward the earth). Let's call the other end the "high end". And, let's assume that the spacecraft is in a circular orbit.
In the same orbit as the spacecraft, place a cigar-shaped column of loose chalk pieces, distributed from a "low end" to a "high end" of the column. The chalk pieces are all initially moving at the same velocities in their orbits as the corresponding parts of the spacecraft.
Now push the "start" button and watch what happens. The spacecraft keeps going in its circular orbit. The column of chalk pieces, however, begins to stretch.
The pieces at the high end are moving too fast to be in a circular orbit at their initial distance from Earth, so they initially move gradually a bit farther from the Earth, losing a bit of speed, until they're halfway around the Earth, at which point they begin to drop gradually closer to the Earth, gaining speed, coming back to their starting point moving at the same initial velocity. Their orbit is slightly ellipsoid rather than circular, and is longer than the spacecraft's orbit. It is also moving on the average slightly slower than the spacecraft.
The pieces at the low end are moving too slow to be in a circular orbit, so they begin to drop closer to the Earth, gaining speed. Halfway around the Earth they reach a maximum speed then start rising while losing speed, and eventually come back to their starting point at the same initial velocity. Their orbit is slightly ellipsoid, too, but is shorter than the spacecraft's orbit; and they move on the average slightly faster than the spacecraft.
So, the low end pieces get back to their starting positions before the high end pieces do. The initial cigar-shaped distribution of chalk pieces is therefore stretched, with the low end pulled forward (in the direction of orbital velocity) and the high end pulled backward.