Does a typical floor pump have an integrated one-way valve? In order for a floor pump to work correctly, something has to make sure that when we pull the pump piston up, the air from the tire does not escape back into the pump. For that there should be a one-way valve between the tube and the pump interior. 

My questions is: does the pump itself have its own integrated one-way valve (installed in the pump head, in the pump body or in the hose)? Or does it rely on the existing tube valve (Presta in my case) to keep the air trapped inside the tube between pump strokes?

In other words, when I use a floor pump to inflate my bicycle tires, is the Presta valve on the wheel supposed to open and close with each stroke of the pump? Or is it permitted to stay open at all times (pressed down by pump head), while an integrated valve in the pump provides the one-way functionality?
There is a valve in the floor pump, yes, so it is supposed to press in the presta valve the whole time it is attached to it.

I too have seen the problem you refer to in the comment on @Useless's answer - diagnosis has always shown this to be down to the presta valve either catching on the side of the pump fitting or just being stiff and refusing to budge.

A simple solution I use is to give it a quick press with my finger before using the pump, just to ensure it moves cleanly and isn't stuck.