Deep underground environments are ideal for low background searches due to the attenuation of cosmic rays by passage through the earth. However, they are affected by backgrounds from $\gamma$-rays emitted by $^{40}$K and the $^{238}$U and $^{232}$Th decay chains in the surrounding rock. The LUX-ZEPLIN (LZ) experiment will search for dark matter particle interactions with a liquid xenon TPC located within the Davis campus at the Sanford Underground Research Facility, Lead, South Dakota, at the 4,850-foot level. In order to characterise the cavern background, in-situ $\gamma$-ray measurements were taken with a sodium iodide detector in various locations and with lead shielding. The integral count rates (0--3300~keV) varied from 596~H...