Let $S$ be the spectrum of a discrete valuation ring $R$ with generic point $\eta ={\rm Spec} K$ and closed point $s = {\rm Spec} K$. Let $\mathcal X$ be a smooth $S$-scheme with generic fiber $X_\eta $, and closed fiber $X_s$. We construct a specialization map of Zariski-$K$-cohomology groups $f : H^1(X_\eta , {\mathcal K}_2) \rightarrow H^1 (X_s, {\mathcal K}_2)$, which depends on the choice of a uniforinizing element $\pi \in R$. Then we show that $f$ is compatible with the natural reduction map from $H^1({\mathcal X},{\mathcal K}_2)$ to $H^1(X_s,{\mathcal K}_2)$. This observation is exploited to prove the following THEOREM. - The notations are as above, in particular let $K$ be a local field of characteristic zero, which is unramified over ${\mathbb Q}_p$ ; let $X_n$, be a smooth projective variety with ordinary good reduction, such that ${\rm dim} X_\eta \leq p - 2$ and $X_s$ is ordinary.Then, if the condition ${\rm Pic}(X_{\bar s})(p) \equiv 0$ is satisfied, the map $f : H^1 (X_{\bar \eta }, {\mathcal K}_2)\rightarrow H^1 (X_{\bar s},{\mathcal K}_2),$ induced by f by passing to the geometric fibers, is surjective on the $p$-primary torsion groups. The proof combines results of Bloch and Kato on $p$-adic étale cohomology in the case of ordinary reduction with assertions of Suslin, Lichtenbaum, Colliot-Thélène and Raskind on Zariski-$K$-cohomology, especially in characteristic $p$.