  We introduce a new interacting particle system on $\mathbb{Z}$, \emph{slowed
$t$-TASEP}. It may be viewed as a $q$-TASEP with additional position-dependent
slowing of jump rates depending on a parameter $t$, which leads to discrete and
nonuniversal asymptotics at large time. If on the other hand $t \to 1$ as
$\text{time} \to \infty$, we prove (1) a law of large numbers for particle
positions, (2) a central limit theorem, with convergence to the fixed-time
Gaussian marginal of a stationary solution to SDEs derived from the particle
jump rates, and (3) a bulk limit to a certain explicit stationary Gaussian
process on $\mathbb{R}$, with scaling exponents characteristic of the
Edwards-Wilkinson universality class in $(1+1)$ dimensions. The proofs relate
slowed $t$-TASEP to a certain Hall-Littlewood process, and use contour integral
formulas for observables of this process. Unlike most previously studied
Macdonald processes, this one involves only local interactions, resulting in
asymptotics characteristic of $(1+1)$-dimensional rather than
$(2+1)$-dimensional systems.
