We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if t is an integer different from 0, 1 or -1 and if A < ℤ/pℤ is not too large (with respect to p), then | A + t A| is significantly larger than 2|A| (unless |t| = 3). In the important case |t| = 2, we obtain for instance | A+ tA| ≥ 2.08 |A|−2.