We study a $q$-deformation for the crossed product of the symmetric group $S_n$ with the Clifford algebra on $n$ anticommuting generators. We obtain a $q$-version of the projective analogue for the classical Young symmetrizer introduced by the second author. Our main tool is an analogue of the Hecke algebra of complex-valued functions on the group $\mbox{GL}_n$ over a $p$-adic field relative to the Iwahori subgroup.