  We investigate whether non-metrizable manifolds in various classes can be
homotopy equivalent to a CW-complex (in short: heCWc), and in particular
contractible. We show that a non-metrizable manifold cannot be heCWc if it has
one of the following properties: it contains a countably compact non-compact
subspace; it contains a copy of an $\omega_1$-compact subset of an
$\omega_1$-tree; it contains a non-Lindel\"of closed subspace functionally
narrow in it. (These results hold for more general spaces than just manifolds.)
We also show that the positive part of the tangent bundle of the long ray is
not heCWc (for any smoothing). These theorems follow from stabilization
properties of real valued maps. On a more geometric side, we also show that the
Pr\"ufer surface, which has been shown to be contractible long ago, has an open
submanifold which is not heCWc. On the other end of the spectrum, we show that
there is a non-metrizable contractible Type I surface.
