The closure \(\mathcal L\) of the Ornstein--Uhlenbeck operator has spectral resolution \[ {\mathcal L}f=\sum_{n=0}^\infty n P_n f, \] where \(P_n\) is the orthogonal projection onto the linear span of Hermite polynomials of degree \(n\) in \(d\) variables. For a given function \(M\) defined in \( \mathbb R^+\), define the operator \(M({\mathcal L})f=\sum_{n=0}^\infty M(n) P_n f\). Then \(M\) is said to be an \(L^p(\gamma)\) spectral multiplier if \(M({\mathcal L})\) extends to a bounded operator on \(L^p(\gamma)\) (\(\gamma\) being the Gauss measure on \( \mathbb R^d\)). In this paper, the authors study the problem of finding conditions on the \(L^p(\gamma)\) spectral multipliers \(M\) that force \(M\) to extend to a holomorphic function in some sector containing the positive real line. In fact, they prove that if \(10} | | M(t{\mathcal L})| | _{L^p(\gamma)}<\infty,\) then \(M\) extends to a holomorphic function in the sector \(\{z\in{\mathbb C}: | \arg\;z| <\arcsin | 2/p-1| \}\). This type of result is a continuation of some previous results in [J. Funct. Anal. 183, 413--450 (2001; Zbl 0995.47010)]. Related results for the case \(p=1\) are also presented.