  We present the first measurements of long-range angular correlations and the
transverse momentum dependence of elliptic flow $v_2$ in high-multiplicity
$p$$+$Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV. A comparison of these
results with previous measurements in high-multiplicity $d$$+$Au and $^3{\rm
He}$$+$Au collisions demonstrates a relation between $v_2$ and the initial
collision eccentricity $\varepsilon_2$, suggesting that the observed
momentum-space azimuthal anisotropies in these small systems have a collective
origin and reflect the initial geometry. Good agreement is observed between the
measured $v_2$ and hydrodynamic calculations for all systems, and an argument
disfavoring theoretical explanations based on momentum-space domain
correlations is presented. The set of measurements presented here allows us to
leverage the distinct intrinsic geometry of each of these systems to
distinguish between different theoretical descriptions of the long-range
correlations observed in small collision systems.
