Two people remain in a critical condition after being found unconscious on a bench near Superdrug in the Maltings shopping area on Sunday afternoon (5th March).
Temporary Assistant Chief Constable Craig Holden of Wiltshire Police explained: "We are conducting some extensive enquiries to determine exactly what led to these two people falling unconscious and clarify whether or not any criminal activity has happened".
"However, I must emphasize that we retain an open mind, and that we continue to review this position".
A former Russian spy is in a critical condition in a British hospital after "suspected exposure to an unknown substance", according to United Kingdom media. He was later swapped and granted refuge in Britain as part of a Cold War style spy exchange in Vienna in 2010.
Freya Church told the BBC it looked like the two people had taken "something quite strong".
The hospital is not appealing for extra staff to come in yet.
A major incident has been declared at Salisbury District Hospital but patients are advised to attend appointments as normal unless advised otherwise.
A public inquiry concluded in 2016 that the killing of Mr Litvinenko, an outspoken critic of Vladimir Putin, had "probably" been carried out with the approval of the Russian president.
She said: "On the bench there was a couple, an older guy and a younger girl".
Salisbury District Hospital's A&E is closed due to an ongoing "major incident".
Sergei Skripal, once a colonel in Russia's GRU military intelligence service, was convicted in Russian Federation of treason in 2006 but exchanged as part of a Cold War-style spy swap in 2010. She was sort of leant-in on him, it looked like she'd passed out maybe.
In 2006, Russian dissident Alexander Litvinenko died after drinking tea laced with radioactive polonium-210 at a London hotel.
He was one of four prisoners released by Moscow in exchange for 10 USA spies in 2010. One of the victims is a former Russian spy, Britain's national broadcaster said.