  For a smooth rigid space $X$ over a complete algebraically closed extension
of $\mathbb{Q}_p$, we investigate how the $v$-Picard group of the associated
diamond differs from the analytic Picard group of $X$. To this end, we
construct an injective "Hodge-Tate logarithm"
\[\mathrm{Pic}_v(X)/\mathrm{Pic}_{\mathrm{an}}(X)\hookrightarrow
H^0(X,\Omega_X^1)(-1)\] and show that this is an isomorphism if $X$ is proper
or one-dimensional. In contrast, we show that for the affine space
$\mathbb{A}^n$, the image consists precisely of the closed differentials.
  It follows that up to a splitting, $v$-line bundles may be interpreted as
Higgs bundles. For proper $X$, we show that this realises the $p$-adic Simpson
correspondence in rank $1$.
