Could we have a position-independent picture for quantum fields?

Here’s what I understood. In quantum mechanics, there is two equivalent of treating time:

1. Scrödinger’s picture where operators are time-independent and states are time-dependent.

2. Heisenberg’s picture where operators are time-dependent and states are time-independent.

In both of these pictures:

• time is seen as an external parameter on which depend observables or states (depending of the choice of formalism).

• position is an observable, so there’s a position operator $hat{boldsymbol{X}}$.

As explained in Srednicki’s book, this difference in the treatment of time and position is why QM is hardly adaptable to special relativity, where time and space are treated on the same foot. One option is then to downgrade position to the status of "external variable". This gives raise to quantum fields wich are spacetime-dependent operators $hat{phi}(t,overrightarrow{x})$.

Now in QFT, we stil have Heisenberg’s and Scrödinger’s picture for the external variable $t$. But since position $overrightarrow{x}$ is now also an external parameter, could we also have an equivalent way of seeing fields as position independent and states as position-dependent?

If so, is this possible to work in this picture and in Schrödinger’s picture at same time, such that operators would be spacetime-independent?