Let R be a discrete valuation ring of mixed characteristics (0,p), with finite residue field k and fraction field K, let k′ be a finite extension of k, and let X be a regular, proper and flat R-scheme, with generic fibre XK and special fibre Xk. Assume that XK is geometrically connected and of Hodge type ≥1 in positive degrees. Then we show that the number of k′-rational points of X satisfies the congruence |X(k′)|≡1 mod |k′|. We deduce such congruences from a vanishing theorem for the Witt cohomology groups Hq(Xk,WXk,ℚ) for q>0. In our proof of this last result, a key step is the construction of a trace morphism between the Witt cohomologies of the special fibres of two flat regular R-schemes X and Y of the same dimension, defined by a surjective projective morphism f:Y→X.