  We consider the iteration of quasiregular maps of transcendental type from
$\mathbb{R}^d$ to $\mathbb{R}^d$. We give a bound on the rate at which the
iterates of such a map can escape to infinity in a periodic component of the
quasi-Fatou set. We give examples which show that this result is best possible.
Under an additional hypothesis, which is satisfied by all uniformly
quasiregular maps, this bound can be improved to be the same as those in a
Baker domain of a transcendental entire function.
  We construct a quasiregular map of transcendental type from $\mathbb{R}^3$ to
$\mathbb{R}^3$ with a periodic domain in which all iterates tend locally
uniformly to infinity. This is the first example of such behaviour in a
dimension greater than two.
  Our construction uses a general result regarding the extension of biLipschitz
maps. In addition, we show that there is a quasiregular map of transcendental
type from $\mathbb{R}^3$ to $\mathbb{R}^3$ which is equal to the identity map
in a half-space.
