  We will construct the Lusztig form for the quantum loop algebra of
$\mathfrak{gl}_n$ by proving the conjecture \cite[3.8.6]{DDF} and establish
partially the Schur--Weyl duality at the integral level in this case. We will
also investigate the integral form of the modified quantum affine
$\mathfrak{gl}_n$ by introducing an affine stabilisation property and will lift
the canonical bases from affine quantum Schur algebras to a canonical basis for
this integral form. As an application of our theory, we will also discuss the
integral form of the modified extended quantum affine $\mathfrak{sl}_n$ and
construct its canonical basis to verify a conjecture of Lusztig in this case.
