  Specific heat, elastic neutron scattering, and muon spin rotation ($\mu$SR)
experiments have been carried out on a well-characterized sample of "stuffed"
(Pr-rich) Pr$_{2+x}$Ir$_{2-x}$O$_{7-\delta}$. Elastic neutron scattering shows
the onset of long-range spin-ice "2-in/2-out" magnetic order at $T_M = 0.93$ K,
with an ordered moment of 1.7(1)$\mu_\mathrm{B}$/Pr ion at low temperatures.
Approximate lower bounds on the correlation length and correlation time in the
ordered state are 170 \AA\ and 0.7 ns, respectively. $\mu$SR experiments yield
an upper bound 2.6(7) mT on the local field $B_\mathrm{loc}^{4f}$ at the muon
site, which is nearly two orders of magnitude smaller than the expected dipolar
field for long-range spin-ice ordering of 1.7$\mu_B$ moments (120--270 mT,
depending on muon site). This shortfall is due in part to splitting of the
non-Kramers crystal-field ground-state doublets of near-neighbor Pr$^{3+}$ ions
by the $\mu^+$-induced lattice distortion. For this to be the only effect,
however, $\sim$160 Pr moments out to a distance of $\sim$14 \AA\ must be
suppressed. An alternative scenario, which is consistent with the observed
reduced nuclear hyperfine Schottky anomaly in the specific heat, invokes slow
correlated Pr-moment fluctuations in the ordered state that average
$B_\mathrm{loc}^{4f}$ on the $\mu$SR time scale (${\sim}10^{-7}$ s), but are
static on the time scale of the elastic neutron scattering experiments
(${\sim}10^{-9}$ s). In this picture the dynamic muon relaxation suggests a
Pr$^{3+}$ $4f$ correlation time of a few nanoseconds, which should be
observable in a neutron spin echo experiment.
