Let $X$ be a smooth affine variety over a finite field of characteristic $p$. The Dwork-Monsky trace formula is a fundamental tool in understanding the $L$-functions of $p$-adic representations of $\pi_1(X)$. We extend this result to the study of representations valued in a higher-dimensional local ring $R$. The special case $R=\mathbb{Z}_p[[T]]$ arises naturally in the study of \'etale $\mathbb{Z}_p$-towers over $X$. Time permitting, we discuss some spectral-halo type results and conjectures describing the $p$-adic variation of slopes in certain $\mathbb{Z}_p$-towers.