  We present global VLBI observations of the ~8100 km s$^{-1}$ HI absorption
feature detected toward the strong radio nucleus 3C 84 of NGC 1275 in the
Perseus cluster (${V_{\rm sys}\sim}$5200 km s$^{-1}$). The observations were
obtained using the Very Long Baseline Array (VLBA), the phased Very Large Array
(VLA), and three stations of the European VLBI Network (EVN). Our results
provide the first high dynamic range image of this feature at high spectral and
spatial resolutions. We detect six distinct absorption peaks with optical
depths ranging from 0.1 to 0.45, and multiple weak features with $\tau \leq$
0.1. The compactness of the background radio source 3C 84, which has a linear
extent of only about 16 pc, limits the conclusions that can be drawn as to the
nature of the intervening object, which must be falling toward the center of
the Perseus cluster at ~3000 km s$^{-1}$. However, the detected absorption
peaks indicate the existence of several HI clouds with velocity differences and
widths similar to those seen in Galactic neutral hydrogen absorption and
similar to some of the HI absorption seen in the Large Magellanic Cloud. The
most prominent HI clouds extend from 12 to 30 mas (milliarcseconds) on the
plane of the sky. The derived HI column densities, assuming $T_{\rm s}=50$ K,
range over $(0.35-3.8)\times 10^{20}$ cm$^{-2}$, and the implied volume
densities range between 1.4 and 11 cm$^{-3}$.
