We study the role of Banach space geometry in deterministic and stochastic maximal regularity. We first construct an example showing that the UMD assumption in Weis' characterisation of maximal Lp-regularity in terms of R-sectoriality cannot be omitted. Combining this construction with an equivalence between stochastic maximal regularity and deterministic maximal regularity on the 2-concavification of the underlying space, we obtain an operator on a UMD Banach function space of type 2 that has a bounded H∞
Nearby in the stack
-calculus of angle zero, but fails stochastic maximal
Lp
-regularity (SMR
p
) for every
p∈[2,∞)
. Motivated by this example, we study the Banach space geometry hypothesis underlying SMR
p
more closely. This is an
R
-boundedness condition
(Sp)
for stochastic convolution operators. For UMD spaces
X
of type
2
, we show that this condition is not only sufficient, but also necessary for two canonical test operators: a diagonal multiplier on a Rademacher space and, for
q>2
, the Laplacian on
Lq(Rd;X)
. Finally, we prove that its interval-kernel and exponential-kernel formulations are equivalent and that, at the endpoint