  The clustering of local extrema will be exploited to examine Gaussianity,
asymmetry, and the footprint of the cosmic-string network on the CMB observed
by Planck. The number density of local extrema ($n_{\rm pk}$ for peak and
$n_{\rm tr}$ for trough) and sharp clipping ($n_{\rm pix}$) statistics support
the Gaussianity hypothesis for all component separations. However, the pixel at
the threshold reveals a more consistent treatment with respect to end-to-end
simulations. A very tiny deviation from associated simulations in the context
of trough density, in the threshold range $\theta\in [-2-0]$ for NILC and CR
component separations, are detected. The unweighted two-point correlation
function, of the local extrema, illustrates good consistency between different
component separations and corresponding Gaussian simulations for almost all
available thresholds. However, for high thresholds, a small deficit in the
clustering of peaks is observed with respect to the Planck fiducial
$\Lambda$CDM model. To put a significant constraint on the amplitude of the
mass function based on the value of $\Psi$ around the Doppler peak
($\theta\approx 70-75$ arcmin), we should consider $\vartheta\lesssim 0.0$. The
scale-independent bias factors for the peak above a threshold for large
separation angle and high threshold level are in agreement with the value
expected for a pure Gaussian CMB. Applying the $n_{\rm pk}$, $n_{\rm tr}$,
$\Psi_{\rm pk-pk}$ and $\Psi_{\rm tr-tr}$ measures on the tessellated CMB map
with patches of $7.5^2$ deg$^2$ size prove statistical isotropy in the Planck
maps. The peak clustering analysis puts the upper bound on the cosmic-string
tension, $G\mu^{(\rm up)} \lesssim 5.59\times 10^{-7}$, in SMICA.
