  Let $F$ be a field of characteristic 2 and let $X$ be a smooth projective
quadric of dimension $\ge 1$ over $F$. We study the unramified cohomology
groups with 2-primary torsion coefficients of $X$ in degrees 2 and 3. We
determine completely the kernel and the cokernel of the natural map from the
cohomology of $F$ to the unramified cohomology of $X$. This extends the results
in characteristic different from 2 obtained by Kahn, Rost and Sujatha in the
nineteen-nineties.
