Exponential $C^0$ decay and $C^\infty$ convergence give exponential $C^\infty$ decay on a half-cylinder

Let $F$ be a real normed space and $f:\mathbb{R}\times\mathbb{R}\to F$. Suppose $f$ tends to $0$ in $C^\infty$ as $s\to+\infty$: $f$ is $C^\infty$ on $(s_1,\infty)\times\mathbb{R}$ for some $s_1$, and for every $m$ the $m$-th derivative $D^mf(s,t)$ tends to $0$ uniformly in $t$. Suppose moreover that $\|f(s,t)\|\le Ce^{-\delta s}$ for all $s\ge s_0$ and all $t$. Then for every $b<\delta$ the function $f$ decays like $e^{-bs}$ in $C^\infty$: for every $m$ there are $C_m$ and $s_m$ with $\|D^mf(s,t)\|\le C_me^{-bs}$ for $s\ge s_m$ and all $t$.

This upgrades exponential $C^0$ decay to exponential $C^\infty$ decay at any slightly smaller rate, uniform over all derivative orders, for maps on a cylindrical end that are already known to converge in $C^\infty$.

Formalization Note The hypothesis is TendstoZeroCInfty f and the conclusion is ∀ b < δ, ExpDecayCInfty f b, with derivatives iteratedFDeriv ℝ m f.