  In 1982, Gessel showed that the Ap\'ery numbers associated to the
irrationality of $\zeta(3)$ satisfy Lucas congruences. Our main result is to
prove corresponding congruences for all sporadic Ap\'ery-like sequences. In
several cases, we are able to employ approaches due to McIntosh, Samol--van
Straten and Rowland--Yassawi to establish these congruences. However, for the
sequences often labeled $s_{18}$ and $(\eta)$ we require a finer analysis.
  As an application, we investigate modulo which numbers these sequences are
periodic. In particular, we show that the Almkvist--Zudilin numbers are
periodic modulo $8$, a special property which they share with the Ap\'ery
numbers. We also investigate primes which do not divide any term of a given
Ap\'ery-like sequence.
