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We consider self-adjoint semigroups $T_t = \exp(-tA)$ acting on $L^2(\Omega)$and satisfying (generalised) Gaussian estimates, where $\Omega$ is a metricmeasure space of homogeneous type of dimension $d$. The aim of the article isto show that $A \otimes \mathrm{Id}_Y$ admits a H\"ormander type$\mathcal{H}^\beta_2$ functional calculus on $L^p(\Omega;Y)$ where $Y$ is a UMDlattice, thus extending the well-known H\"ormander calculus of $A$ on$L^p(\Omega)$. We show that if $T_t$ is lattice positive (or merely admits an$H^\infty$ calculus on $L^p(\Omega;Y)$) then this is indeed the case. Here thederivation exponent has to satisfy $\beta > \alpha \cdot d + \frac12$, where$\alpha \in (0,1)$ depends on $p$, and on convexity and concavity exponents of$Y$. A part of the proof is the new result that the Hardy-Littlewood maximaloperator is bounded on $L^p(\Omega;Y)$. Moreover, our spectral multiplierssatisfy square function estimates in $L^p(\Omega;Y)$. In a variant, we showthat if $e^{itA}$ satisfies a dispersive $L^1(\Omega) \to L^\infty(\Omega)$estimate, then $\beta > \frac{d+1}{2}$ above is admissible independent ofconvexity and concavity of $Y$. Finally, we illustrate these results in avariety of examples.

H\"ormander functional calculus on UMD lattice valued $L^p$ spaces under generalised Gaussian estimates