For $p \in [1,\infty]$, consider the sequence space $\ell^{p}(\mathbb{N})$ consisting of real sequences $x = (x_{i})_{i =1}^{\infty}$ with norm $\lVert x \rVert_{p} = \left(\sum_{i=1}^{\infty}|x_{i}|^{p} \right)^{1/p}$ for $p < \infty$ and $\lVert x \rVert_{\infty} = \sup_{i}|x_{i}|$.
Let $A$ be a bounded linear operator on $\ell^{p}(\mathbb{N})$ for some $p$. Let $u: [0,\infty) \to \ell^{p}(\mathbb{N})$, and write $u(t) = (u_{1}(t), u_{2}(t),\dots)$. Suppose that each coordinate function $u_{i}: [0,\infty) \to \mathbb{R}$ is differentiable on $(0, \infty)$, right-differentiable at $0$, and satisfies
$$u_{i}'(t) = (Au(t))_{i}$$
for all $t \geq 0$. My question: is it the case that $$\lim_{h \to 0}\left \lVert \frac{u(t+h) - u(t)}{h} - Au(t) \right \rVert_{p} = 0$$ for all $t$? If not, is there a reasonable counterexample?