We consider self-adjoint semigroups Tt = exp(−tA) acting on L 2 (Ω) and satisfying (generalised) Gaussian estimates, where Ω is a metric measure space of homogeneous type of dimension d. The aim of the article is to show that A ⊗ IdY admits a Hörmander type H β 2 functional calculus on L p (Ω; Y) where Y is a UMD lattice, thus extending the well-known Hörmander calculus of A on L p (Ω). We show that if Tt is lattice positive (or merely admits an H ∞ calculus on L p (Ω; Y)) then this is indeed the case. Here the derivation exponent has to satisfy β > α · d + 1 2 , where α ∈ (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 maximal operator is bounded on L p (Ω; Y). Moreover, our spectral multipliers satisfy square function estimates in L p (Ω; Y). In a variant, we show that if e itA satisfies a dispersive L 1 (Ω) → L ∞ (Ω) estimate, then β > d+1 2 above is admissible independent of convexity and concavity of Y. Finally, we illustrate these results in a variety of examples.