  We study Picard groups in profinite perfectoid limits of rigid spaces in the
case of good reduction. Our main result is that for sufficiently large covers,
these can be described in terms of the special fibre: For example, our
description implies that for a supersingular variety $X$ of good reduction
$\bar{X}$ over a perfectoid field of characteristic $p$,
  \[\mathrm{Pic}(X^{perf})=\mathrm{Pic}(\bar{X})[1/p].\] Via Raynaud
uniformisation, it moreover allows us to describe Picard groups of pro-\'etale
covers of abeloid varieties, which has applications to pro-\'etale
uniformisation.
  We use our results to answer some general questions about Picard groups and
Picard functors of perfectoid spaces, e.g. we show that these are not always
$p$-divisible. Along the way we construct a "multiplicative Hodge--Tate
spectral sequence" for $\mathcal{O}^\times$.
  This is part III of a series on line bundles on diamonds.
