  The sequence $A(n)_{n \geq 0}$ of Ap\'ery numbers can be interpolated to
$\mathbb{C}$ by an entire function. We give a formula for the Taylor
coefficients of this function, centered at the origin, as a $\mathbb{Z}$-linear
combination of multiple zeta values. We then show that for integers $n$ whose
base-$p$ digits belong to a certain set, $A(n)$ satisfies a Lucas congruence
modulo $p^2$.
