I think the majority, if not all, easily available bilge pumps are centrifugal pumps and as such they will not totally drain the bilge, always leaving a small amount of water behind. Even if it is set down in a recess, it will still leave water in the recess. So if there is not much water in the bilge the pump may be fine, but I would invert it and see if you can see the impeller which may be clogged, that would stop it pumping.
It is also fairly normal for the last "pipe full" of water to drain back into the bilge when the pump is turned off. A non-return valve close to the pump would go a long way to prevent this BUT bilges are not the cleanest of things so such a valve may clog & reduce/prevent pumping, or just jamb open, so the water still drains back.
Without more info, I am not convinced anything is wrong with your system because it sounds pretty normal to me.
If you fitted something with a rubber impeller, like a Jabsco Water Puppy (horribly expensive) the impeller would retain the water in the pipe until the impeller broke or wore, so it is not really a solution.