  In this paper we compute new values of Iwahori Whittaker functions on
$n$-fold metaplectic covers $\widetilde{G}$ of $\mathbf{G}(F)$ with
$\mathbf{G}$ a split reductive group over a non-archimedean local field $F$.
For every Iwahori Whittaker function $\phi$, and for every $g\in\widetilde{G}$,
we evaluate $\phi(g)$ by recurrence relations over the Weyl group using novel
"vector Demazure-Whittaker operators." The general formula and strategy of
proof are inspired by ideas appearing in the theory of integrable systems.
Specializing to the case of $\mathbf{G} = \mathbf{GL}_r$, we construct a
solvable lattice model of a new type associated with the quantum affine super
group $U_q(\widehat{\mathfrak{gl}}(r|n))$ and prove that its partition function
equals $\phi(g)$. To prove this equality we match the recurrence relations on
the lattice model side (obtained from the Yang-Baxter equation) to the
recurrence relations for $\phi(g)$ derived by using the representation theory
of $\widetilde{G}$. Remarkably, there is a bijection between the boundary data
specifying the partition function and the data determining all values of the
Whittaker functions.
