  We present a muon spin relaxation study on the Ising pyrochlore
Nd$_2$Zr$_2$O$_7$ which develops an "all-in-all-out" magnetic order below
0.4~K. At 20~mK far below the ordering transition temperature, the zero-field
muon spin relaxation spectra show no static features and can be well described
by a dynamical Gaussian-broadened Gaussian Kubo-Toyabe function indicating
strong fluctuations of the ordered state. The spectra of the paramagnetic state
(below 4.2~K) reveal anomalously slow paramagnetic spin dynamics and show only
small difference with the spectra of the ordered state. We find that the
fluctuation rate decreases with decreasing temperature and becomes nearly
temperature independent below the transition temperature indicating persistent
slow spin dynamics in the ground state. The field distribution width shows a
small but sudden increase at the transition temperature and then becomes almost
constant. The spectra in applied longitudinal fields are well fitted by the
conventional dynamical Gaussian Kubo-Toyabe function, which further supports
the dynamical nature of the ground state. The fluctuation rate shows a peak as
a function of external field which is associated with a field-induced spin-flip
transition. The strong dynamics in the ordered state are attributed to the
transverse coupling of the Ising spins introduced by the multipole
interactions.
