Additive power series in positive characteristic

Question

Let $K$ be a field of positive characteristic $p$. Consider a power series $F\in K[[X_1,\dots,X_n]]$ that is additive, that is, $$F(X_1+Y_1,\dots,X_n+Y_n)=F(X_1,\dots,X_n)+F(Y_1,\dots,Y_n)\in K[[X_1,\dots,X_n,Y_1,\dots,Y_n]].$$ Is it true that there exists $n$ additive power series $F_1,\dots,F_n$ ($F_i\in K[[X_i]]$) such that $F(X_1,\dots,X_n)=F_1(X_1)+\dots+F_n(X_n)$?