Dobrushin (1972) showed that the interface of a 3D Ising model with minus boundary conditions above the $xy$-plane and plus below is rigid (has $O(1)$-fluctuations) at every sufficiently low temperature. Since then, basic features of this interface -- such as the asymptotics of its maximum -- were only identified in more tractable random surface models that approximate the Ising interface at low temperatures, e.g., for the (2+1)D Solid-On-Solid model. Here we study the large deviations of the interface of the 3D Ising model in a cube of side-length $n$ with Dobrushin's boundary conditions, and in particular obtain a law of large numbers for $M_n$, its maximum: if the inverse-temperature $β$ is large enough, then $M_n / \log n \to 2/α_β$ as $n\to\infty$, in probability, where $α_β$ is given by a large deviation rate in infinite volume. We further show that, on the large deviation event that the interface connects the origin to height $h$, it consists of a 1D spine that behaves like a random walk, in that it decomposes into a linear (in $h$) number of asymptotically-stationary weakly-dependent increments that have exponential tails. As the number $T$ of increments diverges, properties of the interface such as its surface area, volume, and the location of its tip, all obey CLTs with variances linear in $T$. These results generalize to every dimension $d\geq 3$.