Error when tensoring with $p$-adic integers

Question

It must be simple but I cannot find the error in the reasoning. Let $p$ be a prime number, we know that $\mathbb Z_p$ is flat over $\mathbb Z$ so that we can take the tensor product of the exact sequence: $$0\rightarrow \mathbb Z\stackrel{p}{\rightarrow}\mathbb Z\rightarrow \mathbb Z/p\mathbb Z\rightarrow 0$$ with $\mathbb Z_p;$ we have the exact sequence:$$0\rightarrow \mathbb Z_p\stackrel{p}{\rightarrow}\mathbb Z_p\rightarrow \mathbb Z/p\mathbb Z\otimes_{\mathbb Z}\mathbb Z_p\rightarrow 0$$ where the last group should be zero. But multiplication by $p$ is not surjective in $\mathbb Z_p$ (the units of $\mathbb Z_p$ have no pre-image).