We show that if $H$ is a group of polynomial growth whose growth rate is at least quadratic, then the $L_p$ compression of the wreath product $\mathbb{Z}\bwr H$ equals \[{\rm max}\left\{\frac{1}{p},\frac{1}{2}\right\}\] . We also show that the $L_p$ compression of $\mathbb{Z}\bwr \mathbb{Z}$ equals \[{\rm max}\left\{\frac{p}{2p-1},\frac23\right\} \] and that the $L_p$ compression of $(\mathbb{Z}\bwr\mathbb{Z})_0$ (the zero section of $\mathbb{Z}\bwr \mathbb{Z}$ , equipped with the metric induced from $\mathbb{Z}\bwr \mathbb{Z}$ ) equals \[{\rm max}\left\{\frac{p+1}{2p},\frac34\right\} \] . The fact that the Hilbert compression exponent of $\mathbb{Z}\bwr\mathbb{Z}$ equals $2/3$ while the Hilbert compression exponent of $(\mathbb{Z}\bwr\mathbb{Z})_0$ equals $3/4$ is used to show that there exists a Lipschitz function $f:(\mathbb{Z}\bwr\mathbb{Z})_0\to L_2$ which cannot be extended to a Lipschitz function defined on all of $\mathbb{Z}\bwr \mathbb{Z}$ .