$p$-divisibility vs tensoring with $\mathbb{Z}_p$.

Question

Suppose $M$ is any $\mathbb{Z}$-module, and $p$ a prime. Are the following statements equivalent?

1. $M \otimes _{\mathbb{Z}} \mathbb{Z}_p = 0$
2. $M$ is $p$-divisible (i,e., for all $\alpha \in M$ there exists $\beta \in M$ such that $p\beta = \alpha$)

$2 \implies 1$ is straightforward, but I don't see the reverse implication (or any counterexample). Can anyone please provide any proof or counterexample for $1 \implies 2$? Or any possible hypothesis when this can be proved.

Edit: To be clear, by $\mathbb{Z}_p$ I mean the quotient $\mathbb{Z}/p\mathbb{Z}$.