Let $ p $ be a prime number. Does there exist a group $ G $ of order $ o(G)=p^p $ such that all elements in $ G $ different from the identity element have order $ p $ ?

Motivation for the question : An example for $ p=2 $ is obvious, the Klein 4-group. 

For $ p=3 $, I came across this very nice example. Let $ G = U\_3( \mathbb{F}\_3 ) $, the group of all upper triangular matrices over $ \mathbb{F}\_3 $, with all entries on the main diagonal equal to $ 1 $. Note that $ o(G) = 27 $ and for any $ M \in G, M \neq I $, we can write $ M=I+N $, $ N \neq 0 $. Note that $ N $ is a strictly upper triangular matrix, hence nilpotent, $ N^3 = 0 $. From this, we get $ M^3 = (I+N)^3 = I + N^3 = I $ and hence all elements other than the identity are of order $ 3 $.

As pointed out below, $ G = (\mathbb{Z}\_p)^p $ does the trick. For $ p \ge 3 $, does there exist an example which is not isomorphic to $ (\mathbb{Z}\_p)^p $?