  We study mapping properties of the centered Hardy--Littlewood maximal
operator $\mathcal{M}$ acting on Lorentz spaces $L^{p,q}(\mathfrak{X})$ in the
context of certain non-doubling metric measure spaces $\mathfrak{X}$. The
special class of spaces for which these properties are very peculiar is
introduced and many examples are given. In particular, for each $p_0, q_0, r_0
\in (1, \infty)$ with $r_0 \geq q_0$ we construct a space $\mathfrak{X}$ for
which the associated operator $\mathcal{M}$ is bounded from
$L^{p_0,q_0}(\mathfrak{X})$ to $L^{p_0,r}(\mathfrak{X})$ if and only if $r \geq
r_0$.
