  For each of the $15$ known sporadic Ap\'ery-like sequences, we prove
congruences modulo $p^2$ that are natural extensions of the Lucas congruences
modulo $p$. This extends a result of Gessel for the numbers used by Ap\'ery in
his proof of the irrationality of $\zeta(3)$. Moreover, we show that each of
these sequences satisfies two-term supercongruences modulo $p^{2r}$. Using
special constant term representations recently discovered by Gorodetsky, we
prove these supercongruences in the two cases that remained previously open.
