Let $\mathbb{F}_p$ be a finite field, $A=\{a_1,\dots,a_k\}\subset\mathbb{F}_p^*$ a $k$-element set, for $k<p$. $\mathfrak{S}_k=$permutation gp.

Question.Is it true there is always a $\pi\in\mathfrak{S}_k$ such that the following are pair-wise distinct in $\mathbb{F}_p$? $$a_{\pi(1)}, \,a_{\pi(1)}+a_{\pi(2)},\,a_{\pi(1)}+a_{\pi(2)}+a_{\pi(3)},\,\dots, \,a_{\pi(1)}+\cdots+a_{\pi(k)}.$$

**EDIT.** Due to Julian Rosen's example, I change $A$ to be a subset of $\mathbb{F}_p^*$ (non-zero elements).