  Several constructions have been given for families of simple braces, but few
examples are known of simple skew braces which are not braces. In this paper,
we exhibit the first example of an infinite family of simple skew braces which
are not braces and which do not arise from nonabelian simple groups. More
precisely, we show that, for any primes $p$, $q$ such that $q$ divides
${(p^p-1)}/{(p-1)}$, there are exactly two simple skew braces (up to
isomorphism) of order $p^p q$.
