  Context. It is well known that from 1645 to 1715 solar activity was notably
low and the number of sunspots was extremely reduced. This epoch is known as
the Maunder Minimum (MM). The study of stars in prolonged activity minima like
the MM could help to shed light on this enigmatic epoch.
  Aims. To explore if the star $\zeta^{2}$ Ret, which belongs to a binary
system, is in a state similar to the MM. We have collected more than 430
spectra acquired between 2000 and 2019 with the HARPS, REOSC, UVES, and FEROS
spectrographs.
  Methods. We performed a detailed long-term activity study of both components
using the Mount Wilson index, which is obtained from the Ca II H\&K lines. To
search for signs of an activity cycle, we analyzed the resulting time-series
with the Generalized Lomb-Scargle and CLEAN periodograms.
  Results. Our spectroscopic analysis shows a high activity level for zet01 Ret
and a significant decrease in the magnetic activity cycle amplitude of
$\zeta^{2}$ Ret. Thus, the activity difference between both components has been
slightly increased ($\Delta$log (RHK) $\sim$0.24 dex), when compared to the
previously reported value. The long series analyzed here allowed us to
recalculate and constrain the period of $\zeta^{2}$ Ret to $\sim$7.9 yr. We
also detected a long-term activity cycle of $\sim$4.2 yr in $\zeta^{1}$ Ret.
  Conclusions. By analogy with the scenario that proposes a weak solar cycle
during the MM, we suggest that activity signatures showed by $\zeta^{2}$ Ret,
i.e., a very low activity level when compared to its stellar companion, a
notably decreasing amplitude ($\sim$47%), and a cyclic behaviour, are possible
evidence that this star could be in a MM state. It is, to our knowledge, the
first MM candidate star detected through a highly discrepant activity behaviour
in a binary system.
