Abstract Differential cross sections of $p$–$\mathrm{^6He}$ elastic scattering were measured in inverse kinematics at an incident energy of 200 $A \mathrm{MeV}$, covering the high momentum transfer region of 1.7–2.7 $\mathrm{fm^{-1}}$. The sensitivity of the elastic scattering at low and high momentum transfers to the density distribution was investigated quantitatively using relativistic impulse approximation calculations. In the high momentum transfer region, where the present data were taken, the differential cross section has an order of magnitude higher sensitivity to the inner part of the $\mathrm{^6He}$ density relative to the peripheral part (15:1). This feature makes the obtained data valuable for the deduction of the inner part of the $\mathrm{^6He}$ density. The data were compared to a set of calculations assuming different proton and neutron density profiles of $\mathrm{^6He}$. The data are well reproduced by the calculation assuming almost the same profiles of proton and neutron densities around the center of $\mathrm{^6He}$, and a proton profile reproducing the known point-proton radius of 1.94 fm. This finding is consistent with the assumption that the $\mathrm{^6He}$ nucleus consists of a rigid $\alpha$-like core with a two-neutron halo.