Do quantum fields need to be operator-valued distributions in non-relativistic quantum field theory?

Question

In relativistic quantum field theory, there is a result saying that if

1. an operator-valued map $\Phi (x)$ satisfying Poincare covariance is a well-defined operator at any point $x \in \mathbb{R}^4$, and
2. the vacuum is the only translationally-invariant state,

then $\Phi (x)$ must be a constant operator. Does an equivalent result hold for non-relativistic quantum field theory?