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.
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