  Semileptonic $B_s \to K \ell \nu$ decays provide an alternative $b$-decay
channel to determine the CKM matrix element $|V_{ub}|$, and to obtain a
$R$-ratio to investigate lepton-flavor-universality violations. Results for the
CKM matrix element may also shed light on the discrepancies seen between
analyses of inclusive or exclusive decays. We calculate the decay form factors
using lattice QCD with domain-wall light quarks and a relativistic $b$-quark.
We analyze data at three lattice spacings with unitary pion masses down to
$268\,\mathrm{MeV}$. Our numerical results are interpolated/extrapolated to
physical quark masses and to the continuum to obtain the vector and scalar form
factors $f_+(q^2)$ and $f_0(q^2)$ with full error budgets at $q^2$ values
spanning the range accessible in our simulations. We provide a possible
explanation of tensions found between results for the form factor from
different lattice collaborations. Model- and truncation-independent
$z$-parameterization fits following a recently proposed Bayesian-inference
approach extend our results to the entire allowed kinematic range. Our results
can be combined with experimental measurements of $B_s \to D_s$ and $B_s\to K$
semileptonic decays to determine $|V_{ub}|=3.8(6)\times 10^{-3}$. The error is
currently dominated by experiment. We compute differential branching fractions
and two types of $R$ ratios, the one commonly used as well as a variant better
suited to test lepton-flavor universality.
