sum-sets in a finite field 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).
 A: I first learned about this problem from Éric Balandraud (in 2013). So I wrote to him a couple of days ago, and he has just sent me an e-mail explaining that the question dates back (at least) to 1971. Éric attributes it to Erdős and Graham: He hasn't provided me with a reference (he is in the desert these days), but I will come back and update this answer as soon as I hear from him again.
Update (March 31, 2017). I've just got a new message from Éric Balandraud. Below is an excerpt.

Here is the reference:

R. Graham, On sums of integers taken from a fixed sequences, Proc.
  Wash. State Univ. Conf. on Number Theory (1971), 22-40.

The question is stated as Conjecture 10 on page 36. It also appears in:

P. Erdős and R. Graham, Old and new problems and results in combinatorial
  number theory, Monographie N28 de L'Enseignement Mathématiques,
  Université de Genève (1980).

The conjecture is stated there on page 95.

A: It took me some effort to find the references you were requesting in your comment, but here they are eventually: 


*

*Ordering subsets of the cyclic group to give distinct partial sums (posted to MO April 2014);

*Geometry, Number Theory and Graph Theory of n-gon, permutation and graph labeling? (posted to MO June 2015);

*"On partial sums in cyclic groups" by D. Archdeacon, J. Dinitz, A. Mattern, and D. Stinson, (J. Combin. Math. Combin. Comput. 98 (2016), 327–342).
As one can see, the problem is around for over 15 years at least.
