We study the baryon--baryon interactions in the strangeness $S=-1$ sector by applying the time-dependent HAL QCD method to the $N_\mathrm{f}=2+1$ gauge configurations generated at the physical point $(m_\pi,m_K)=(137,502)~\mathrm{MeV}$ (HAL-conf-2023). We analyze the $S$-wave $N\Lambda$--$N\Sigma$ coupled channels ($I =1/2$, $^1S_0$ and $^3S_1$) and $N\Sigma$ ($I=3/2$, $^1S_0$ and $^3S_1$) channels. The extracted potentials for $N\Sigma(I=1/2, {}^1S_0)$ and $N\Sigma(I=3/2, {}^3S_1)$ severely suffer from inelastic excited-state contamination at small times and large statistical fluctuations at larger times. We focus on $N\Sigma(I=3/2, {}^3S_1)$ and try to remove the inelastic contamination by extrapolation to $t\to\infty$ under a simple assumption on the time dependence of the potential. We examine the time dependence of the phase shifts and compare them with data from the J-PARC E40 experiment. We find that the extrapolation method seems to work very well in removing the inelastic contamination.