  From observations at low and high redshifts, it is well known that the bulk
of dark matter (DM) has to be stable or at least very long-lived. However, the
possibility that a small fraction of DM is unstable or that all DM decays with
a half-life time ($\tau$) significantly longer than the age of the Universe is
not ruled out. One-body decaying dark matter (DDM) consists of a minimal
extension to the $\Lambda$CDM model. It causes a modification of the cosmic
growth history as well as a suppression of the small-scale clustering signal,
providing interesting consequences regarding the $S_8$ tension, which is the
observed difference in the clustering amplitude between weak-lensing (WL) and
cosmic microwave background (CMB) observations. In this paper, we investigate
models in which a fraction or all DM decays into radiation, focusing on the
long-lived regime, that is, $\tau \gtrsim H_0^{-1}$ ( $H_0^{-1}$ being the
Hubble time). We used WL data from the Kilo-Degree Survey (KiDS) and CMB data
from Planck. First, we confirm that this DDM model cannot alleviate the $S_8$
difference. We then show that the most constraining power for DM decay does not
come from the nonlinear WL data, but from CMB via the integrated Sachs-Wolfe
effect. From the CMB data alone, we obtain constraints of $\tau \geq 288$~Gyr
if all DM is assumed to be unstable, and we show that a maximum fraction of
$f=0.07$ is allowed to decay assuming the half-life time to be comparable to
(or shorter than) one Hubble time. The constraints from the KiDS-1000 WL data
are significantly weaker, $\tau \geq 60$~Gyr and $f<0.34$. Combining the CMB
and WL data does not yield tighter constraints than the CMB alone, except for
short half-life times, for which the maximum allowed fraction becomes $f=0.03$.
All limits are provided at the 95% confidence level.
