Let $X$ be a proper scheme over a henselian discrete valation ring. I have a Nisnevich sheaf $F$ of which has only one stalk at the generic point of $X$ (and all other stalks vanish).

I believe that this implies that $F$ has no cohomology. This is known, for example, if $X$ itself is the spectrum of a henselian discrete valuation ring (see Milne's arithmetic duality theorems).

Has anybody any idea, or a counter-example?