For each $p$ in $[1, \infty)$ let ${\bf E}_p$ denote the closure of the region of holomorphy of the Ornstein–Uhlenbeck semigroup $\{{\cal H}_t : t >0\}$ on $L^p$ with respect to the Gaussian measure. Sharp weak type and strong type estimates are proved for the maximal operator $f \mapsto {{\cal H}^*}_pf=\sup\{\vert {\cal H}_zf\vert :z\in {\bf E}_p\}$ and for a class of related operators. As a consequence, a new and simpler proof of the weak type 1 estimate is given for the maximal operator associated to the Mehler kernel.