Yes, but for reasons independent of its climb speed.
Creatures with a climb speed may move at their turn rate per turn instead of half their turn movement on the walls. Creatures with climb speeds do not gain any explicit benefit from moving above the ceilings, though any creature that can reasonably resize a ceiling can still do so. A creature spends 2 feet on 1 foot of its movement climbing a ceiling, whether or not it has a climb speed.
A creature with a rate of climb can use all or part of its movement
to move on vertical surfaces (it's me who emphasizes). The monster does not need to do more movement to climb. (MM p.8)
Any character or creature that can perform a track and field check and support its own weight can be suspended or suspended from the ceiling, albeit with a track and field check (up to the DM).
… The DM asks for a capacity check when a character or a monster
tries an action … that has a chance to fail. (PHB page 171)