  We show that there is a transcendental meromorphic function with an invariant
Baker domain $U$ such that every singular value of $f$ is a super-attracting
periodic point. This answers a question of Bergweiler from 1993. We also show
that $U$ can be chosen to contain arbitrarily large round annuli, centred at
zero, of definite modulus. This answers a question of Mihaljevi\'c and the
author from 2013, and complements recent work of Bara\'nski et al concerning
this question.
