  Let $R$ be an algebraically closed field and $\ell$ be its characteristic.
Let $G$ be a locally profinite group having a compact open subgroup of
invertible pro-order in $R$. Take $N$ a closed subgroup of $G$ exhausted by
compact subgroups of invertible pro-orders in $R$ and fix a smooth character
$\theta$ of $N$. For $\pi$ an irreducible smooth $R$-representation of $G$
whose matrix coefficients are compactly supported modulo the center (we call it
$Z$-compact), we show that the dimensions $\mathrm{Hom}_{N}(\pi,\theta)$ and
$\mathrm{Hom}_{N}(\pi^\vee,\theta^{-1})$ are equal provided one of the two is
finite. We derive a few applications from this result. First, we prove that any
$G$-intertwiner from $\pi$ to $\mathrm{Ind}_N^G(\theta)$ has image in
$\mathrm{ind}_{ZN}^G(\omega_\pi \theta)$, where $\omega_\pi$ is the central
character of $\pi$, and the Whittaker space of $\pi$ agrees with that of its
Whittaker periods. Second, it applies to quasi-split groups over non
Archimedean local fields of residual characteristic $p \neq \ell$ and where $N$
is the unipotent radical of a Borel subgroup of $G$ together with a generic
character $\theta$. Our equality of dimensions turns out to be a good
replacement for Rodier's crucial use of complex conjugation in the proof of
Whittaker multiplicity at most one for cuspidal representations. Then by a
lifting argument, we recover Rodier's generalization of the Gelfand-Kazhdan
property for $R$-valued $(\theta^{-1}\otimes \theta)$-equivariant distributions
on $G$. This latter fact, together with Rodier's heridity property, which is
valid in our context, leads to the multiplicity at most one of Whittaker
functionals over $R$. We also give other applications, including a
generalization over $R$ of a result for complex representations proved by Chang
Yang and initially conjectured by Dipendra Prasad.
