  Recent LHCb data for $B^-\to J/\psi\Lambda\bar{p}$ show a clear peak
structure at the $\Xi_c\bar{D}$ threshold in the $J/\psi\Lambda$ invariant mass
($M_{J/\psi\Lambda}$) distribution. The LHCb's amplitude analysis identified
the peak with the first hidden-charm pentaquark with strangeness $P_{\psi
s}^\Lambda(4338)$. We conduct a coupled-channel amplitude analysis of the LHCb
data by simultaneously fitting the $M_{J/\psi\Lambda}$, $M_{J/\psi\bar{p}}$,
$M_{\Lambda\bar{p}}$, and $\cos\theta_{K^*}$ distributions. Rather than the
Breit-Wigner fit employed in the LHCb analysis, we consider relevant threshold
effects and a unitary $\Xi_c\bar{D}$-$\Lambda_c\bar{D}_s$ coupled-channel
scattering amplitude from which $P_{\psi s}^\Lambda$ poles are extracted for
the first time. In our default fit, the $P_{\psi s}^\Lambda(4338)$ pole is
almost a $\Xi_c \bar{D}$ bound state at $( 4338.2\pm 1.4)-( 1.9\pm 0.5 )\,i$
MeV. Our default model also fits a large fluctuation at the
$\Lambda_c\bar{D}_s$ threshold, giving a $\Lambda_c\bar{D}_s$ virtual state,
$P_{\psi s}^\Lambda(4255)$, at $4254.7\pm 0.4$ MeV. We also found that the
$P_{\psi s}^\Lambda(4338)$ peak cannot solely be a kinematical effect, and a
nearby pole is needed.
