  For a smooth rigid space $X$ over a perfectoid field extension $K$ of
$\mathbb Q_p$, we investigate how the $v$-Picard group of the associated
diamond $X^\diamondsuit$ differs from the analytic Picard group of $X$. To this
end, we construct a left-exact "Hodge--Tate logarithm" sequence \[0\to
\mathrm{Pic}_{\mathrm{an}}(X)\to \mathrm{Pic}_v(X^\diamondsuit)\to
H^0(X,\Omega_X^1)\{-1\}.\]
  We deduce some analyticity criteria which have applications to $p$-adic
modular forms.
  For algebraically closed $K$, we show that the sequence is also right-exact
if $X$ is proper or one-dimensional. In contrast, we show that for the affine
space $\mathbb A^n$, the image of the Hodge--Tate logarithm 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 use this to construct the $p$-adic Simpson
correspondence of rank one.
