  Consider the 3D Ising model on a box of side length $n$ with minus boundary
conditions above the $xy$-plane and plus boundary conditions below it. At low
temperatures, Dobrushin (1972) showed that the interface separating the
predominantly plus and predominantly minus regions is localized: its height
above a fixed point has exponential tails. Recently, the authors proved a law
of large numbers for the maximum height $M_n$ of this interface: for every
$\beta$ large, $M_n/ \log n\to c_\beta$ in probability as $n\to\infty$.
  Here we show that the laws of the centered maxima $(M_n -
\mathbb{E}[M_n])_{n\geq 1}$ are uniformly tight. Moreover, even though this
sequence does not converge, we prove that it has uniform upper and lower Gumbel
tails (exponential right tails and doubly exponential left tails). Key to the
proof is a sharp (up to $O(1)$ precision) understanding of the surface large
deviations. This includes, in particular, the shape of a pillar that reaches
near-maximum height, even at its base, where the interactions with neighboring
pillars are dominant.
