A stochastic process $X(t)$ is differentiable stochastically, in $L^p$ or almost surely by definition iff:
$$\lim\limits_{\Delta t \rightarrow 0}\frac{X(t+\Delta t)-X(t)}{\Delta t}=X'(t)$$
for each $t$ in the corresponding sense, where $X'(t)$ is another stochastic process (see here ).
Is the Ornstein-Uhlenbeck process differentiable stochastically or in $L^p$ or almost surely? How to prove this?
It isn't -- the OU process is as rough as standard Brownian motion.
In fact, the defining SDE of the OU process shows that the variance of the difference quotient at any given $t$ diverges as $\Delta t \to 0$.