  Let $A$ be a generator of an analytic semigroup having a H{\"o}rmander
functional calculus on $X = L^p(\Omega ,Y)$, where $Y$ is a UMD lattice. Using
methods from Banach space geometry in connection with functional calculus, we
show that for H{\"o}rmander spectral multipliers decaying sufficiently fast at
$\infty$, there holds a maximal estimate $\| \sup_{t \geq 0} |m(tA)f|\,
\|_{L^p(\Omega ,Y)} \lesssim \|f\|_{L^p(\Omega ,Y)}$. We also show square
function estimates $\left\| \left( \sum_k \sup _{t \geq 0} |m_k(tA)f_k|^2
\right)^{\frac12} \right\|_{L^p(\Omega ,Y)} \lesssim \left\| \left( \sum _k
|f_k|^2 \right)^{\frac12} \right\|_{L^p(\Omega ,Y)}$ for suitable families of
spectral multipliers $m_k$, which are even new for the euclidean Laplacian on
scalar valued $L^p(\mathbb{R}^d)$. As corollaries, we obtain maximal estimates
for wave propagators and Bochner--Riesz means. Finally, we illustrate the
results by giving several examples of operators $A$ that admit a H{\"o}rmander
functional calculus on some $L^p(\Omega ,Y)$ and discuss examples of lattices
$Y$ and non-self-adjoint operators $A$ fitting our context.
