  Motivated by questions about $\mathbb{C}_p$-valued Fourier transform on the
locally compact group $(\mathbb{Q}_p^d,+)$, we study invariant norms on the
$p$-adic Schr\"odinger representation of the Heisenberg group. Our main result
is a minimality and rigidity property for norms in a family of invariant norms
parameterized by a Grassmannian. This family is the orbit of the sup norm under
the action of the symplectic group, acting via intertwining operators. We also
prove general fundamental properties of quotients of the universal unitary
completion of cyclic algebraic representations. Combined with the rigidity
property, we are able to show that the completion of the Schr\"odinger
representation in any of the norms in that family satisfies a strong notion of
irreducibility and a strong version of Schur's lemma. Norms that can be formed
as the maximum of a finite number of norms from that family are also studied.
We conclude this paper with a list of open questions.
