  We define an integral form of shifted quantum affine algebras of type $A$ and
construct Poincar\'e-Birkhoff-Witt-Drinfeld bases for them. When the shift is
trivial, our integral form coincides with the RTT integral form. We prove that
these integral forms are closed with respect to the coproduct and shift
homomorphisms. We prove that the homomorphism from our integral form to the
corresponding quantized $K$-theoretic Coulomb branch of a quiver gauge theory
is always surjective. In one particular case we identify this Coulomb branch
with the extended quantum universal enveloping algebra of type $A$. Finally, we
obtain the rational (homological) analogues of the above results (proved
earlier in arXiv:1611.06775, arXiv:1806.07519 via different techniques).
