As requested, I am reposting [my comment][1] as an answer. Note that [Boucksom, Favre, and Jonsson][2] 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][2], 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 $X$:

$\hskip2.25in$[![Blowup of P2 at a point][3]][3]

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][4], 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
$$\operatorname{vol}_{X \vert C}(h+te) =
\begin{cases}
  a+bt & t\le0,\\
  a & t\ge0,
\end{cases}$$
which is not differentiable at $t = 0$ if $b \ne 0$.

  [1]: https://mathoverflow.net/questions/315344/differentiability-of-restricted-volumes-of-big-divisors#comment786826_315344
  [2]: https://doi.org/10.1090/S1056-3911-08-00490-6
  [3]: https://gist.githubusercontent.com/takumim/525247abc6b2e2795c0da5dfe21e3101/raw/8b67fab06ef718a9520bdd231f1e368e965f79a5/BlpP2-ZD.svg?sanitize=true
  [4]: https://doi.org/10.1353/ajm.0.0054