(Sort-of) Converse of the Eisenstein criterion

Can anyone find a proof or counterexample to the following statement:

Let f in Z[x] be an irreducible monic polynomial. Then, there exist an
integer a and a prime p such that f(x + a) = a_0 + a_1 * x + ... + x^n,
where p divides a_i for each i, and p ^ 2 does not divide a_0.

Note that the converse of this statement follows immediately from the
Eisenstein criterion.

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