As requested, I am reposting my comment as an answer. Note that Boucksom, Favre, and Jonsson only state that $X = \operatorname{Bl}_P\mathbf{P}^2$ gives an example of a restricted volume that is not $\mathcal{C}^1$. We will show that their example also gives an example of a non-differentiable restricted volume.
Example (see [Boucksom–Favre–Jonsson, Ex. 4.17]). Let $X$ be the blowup of $\mathbf{P}^2$ at a point, and let $h,e \in N^1(X)$ be the class of the pullback of a line and the exceptional class, respectively. We then have the following decomposition of the effective cone of $X$:
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The positive part $P$ of the Zariski decomposition for a point $\alpha = xh-ye$ is as described in the figure above. By [Ein–Lazarsfeld–Mustaţă–Nakamaye–Popa, Ex. 2.19], the restricted volume for the class $\alpha = xh-ye$ along a curve $C$ with class $ah-be$ can be computed as $$\begin{aligned} \operatorname{vol}_{X \vert C}(\alpha) &= P(\alpha)\cdot(ah-be)\\ &= \begin{cases} ax-by & \alpha \in \operatorname{Nef}^1(X),\\ ax & \alpha \in \operatorname{\overline{Eff}}^1(X) \smallsetminus \operatorname{Nef}^1(X). \end{cases} \end{aligned}$$ For $\alpha = h+te$, we have $$\operatorname{vol}_{X \vert C}(h+te) = \begin{cases} a+bt & t\ge0,\\ a & t\le0, \end{cases}$$ which is not differentiable at $t = 0$ if $b \ne 0$.