= Solution
The berth moves at $0.85c$ in the ship's initial <rest frame>, so it too is subject to <length contraction>:
$$
\boxed{L_b'=\frac{L_b}{\gamma_s}=100(1-0.85^2)+4\sqrt{1-0.85^2}\simeq29.8571\,\mathrm m.}
$$
The ship remains $100\,\mathrm m$ long in this frame. The two length comparisons use different simultaneous endpoint events; this is the same <relativity of simultaneity> that resolves the <ladder paradox>. Physical docking also requires deceleration, whose later effects are outside these initial inertial measurements.
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