This article is devoted to construction of infinitely based series of identities. Such counterexamples in Specht problem are built in any positive characteristics. The main result is the following: Theorem. Let $F$ be any field of characteristic $p$, $q=p^s$, $s>1$. Then the polynomials $R_n$: $$ R_n=[[E,T],T]\prod_{i=1}^n Q(x_i,y_i) ([T,[T,F]][[E,T],T])^{q-1}[T,[T,F]], $$ where $Q(x,y)=x^{p-1}y^{p-1}[x,y]$, generate an infinitely based variety.