We prove that there is an absolute constant c > 0 with the following property: if Z / p Z denotes the group of prime order p, and a subset A ⊂ Z / p Z satisfies 1 < | A | < p / 2 , then for any positive integer m < min { c | A | / ln | A | , p / 8 } there are at most 2 m non-zero elements b ∈ Z / p Z with | ( A + b ) ∖ A | ⩽ m . This (partially) extends onto prime-order groups the result, established earlier by S. Konyagin and the present author for the group of integers. We notice that if A ⊂ Z / p Z is an arithmetic progression and m < | A | < p / 2 , then there are exactly 2 m non-zero elements b ∈ Z / p Z with | ( A + b ) ∖ A | ⩽ m . Furthermore, the bound c | A | / ln | A | is best possible up to the value of the constant c. On the other hand, it is likely that the assumption m < p / 8 can be dropped or substantially relaxed.