Assume there is an $m$-dimensional Brownian motion $(W_t)_{0\le t\le T}$ and a square-integrable, progressively measurable $\mathbb R^{1,m}$-valued stochastic process $(Z_t)_{0 \le t \le T}$ (not necessarily continuous) and we know that the function
$$t \mapsto \int_0^t Z_s \mathrm d W_s$$
is almost surely pathwise continuously differentiable in the classical sense.
Can we then say anything restrictive about $Z$, e.g. that it is equal to $0$?
Thanks for your help!