  Pyrochlore materials are characterized by their hallmark network of
corner-sharing rare-earth tetrahedra, which can produce a wide array of complex
magnetic ground states. Ferromagnetic Ising pyrochlores often obey the
"two-in-two-out" spin ice rules, which can lead to a highly-degenerate spin
structure. Large moment systems, such as Ho$_2$Ti$_2$O$_7$ and
Dy$_2$Ti$_2$O$_7$, tend to host a classical spin ice state with low-temperature
spin freezing and emergent magnetic monopoles. Systems with smaller effective
moments, such as Pr$^{3+}$-based pyrochlores, have been proposed as excellent
candidates for hosting a "quantum spin ice" characterized by entanglement and a
slew of exotic quasiparticle excitations. However, experimental evidence for a
quantum spin ice state has remained elusive. Here, we show that the
low-temperature magnetic properties of Pr$_2$Sn$_2$O$_7$ satisfy several
important criteria for continued consideration as a quantum spin ice. We find
that Pr$_2$Sn$_2$O$_7$ exhibits a partially spin-frozen ground state with a
large volume fraction of dynamic magnetism. Our comprehensive bulk
characterization and neutron scattering measurements enable us to map out the
magnetic field-temperature phase diagram, producing results consistent with
expectations for a ferromagnetic Ising pyrochlore. We identify key hallmarks of
spin ice physics, and show that the application of small magnetic fields
($\mu_0 H_c \sim$0.75T) suppresses the spin ice state and induces a long-range
ordered magnetic structure. Together, our work clarifies the current state of
Pr$_2$Sn$_2$O$_7$ and encourages future studies aimed at exploring the
potential for a quantum spin ice ground state in this system.
