Why my banana sauce is not smooth?
A blender certainly sounds like the easiest method.

If you prefer straining, but your sieve isn't fine enough to filter out the things you don't like, use cheesecloth. You can use enough layers to get a really fine strain, though bananas are thick enough that you'll have to squeeze it through.

The other thing you can try for things like this is a food mill, with the finest disk you have, but it sounds like you don't even want tiny tiny particles, so that may not be good enough for you - and it takes a bit of time and elbow grease.