  We extend the Garkusha-Panin and Voevodsky strict $\mathbb{A}^{1}$-invariance
theorems in motivic homotopy theory to one-dimensional base schemes with
perfect residue fields. In the proof, we utilize the trivial fiber topology and
localization for motivic homotopy categories relative to closed immersions.
This approach allows us to reduce strict $\mathbb{A}^{1}$-invariance to the
case of fields. Furthermore, we compare motivic and Nisnevich local fibrant
replacement functors and study $\mathbb{G}_{m}$-loops.
