Is it possible for an object to have a constant speed but be accelerating?

Yes.  'Acceleration' does NOT mean speeding up.

Acceleration means any change in the speed or direction of motion.
Speeding up, slowing down, or rounding a curve are all accelerations.

Any object that's not moving in a straight line is accelerating, even if
its speed is constant.  

-- Your car is accelerating when you drive 30 mph along a curved road.

-- You're accelerating on a merry-go-round (carousel) or a ferris wheel
when it's speeding up, slowing down, or turning at a constant rate.

-- Your car is accelerating when you see a red light and step on the brakes.

-- You are accelerating when you jog around a circular track at gym.