  Let $\Omega^d$ be the $d$-dimensional Drinfeld symmetric space for a finite
extension $F$ of $\mathbb{Q}_p$. Let $\Sigma^1$ be a geometrically connected
component of the first Drinfeld covering of $\Omega^d$ and let $\mathbb{F}$ be
the residue field of the unique degree $d+1$ unramified extension of $F$. We
show that the natural homomorphism determined by the second Drinfeld covering
from the group of characters of $(\mathbb{F}, +)$ to $\text{Pic}(\Sigma^1)[p]$
is injective. In particular, $\text{Pic}(\Sigma^1)[p] \neq 0$. We also show
that all vector bundles on $\Omega^1$ are trivial, which extends the classical
result that $\text{Pic}(\Omega^1) = 0$.
