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Naturally occurring $\alpha$ emitters with extremely long half-lives are investigated using the latest experimental data. Within the time-dependent perturbation theory, $\alpha$ decay with a rather narrow width is treated as a quasi-stationary problem by dividing the potential between the $\alpha$ particle and daughter nucleus into a stationary part and a perturbation. The experimental $\alpha$ decay half-lives of seven available long-lived $\alpha$ emitters with $T^{\text{total}}_{1/2}>10^{14}$ y are reproduced with a good accuracy. It is also found that the deformation effect should be treated carefully for long-lived nuclei, especially with low $Q_\alpha$ values. Predictions of the $\alpha$ decay half-lives of twenty naturally occurring nuclei are provided, namely, $^{142}$Ce, $^{145,146}$Nd, $^{149}$Sm, $^{156}$Dy, $^{162,164}$Er, $^{168}$Yb, $^{182,183,184,186}$W, $^{187,188,189,190}$Os, $^{192,195}$Pt, and $^{204,206}$Pb. These nuclei are energetically unstable to $\alpha$ decay with low decay energies and extremely long decay half-lives. In particular, the candidates $^{187}$Os and $^{149}$Sm are strongly recommended for future experiments.