When Saruman is killed,
To the dismay of those that stood by, about the body of Saruman a grey mist gathered, and rising slowly to a great height like smoke from a fire, as a pale shrouded figure it loomed over the Hill. For a moment it wavered, looking to the West; but out of the West came a cold wind, and it bent away, and with a sigh dissolved into nothing.
This is almost identical to what happens to Sauron:
'The realm of Sauron is ended!' said Gandalf. 'The Ring-bearer has fulfilled his
Quest.' And as the Captains gazed south to the Land of Mordor, it seemed to them
that, black against the pall of cloud, there rose a huge shape of shadow,
impenetrable, lightning-crowned, filling all the sky. Enormous it reared above the world, and stretched out towards them a vast threatening hand, terrible but impotent: for even as it leaned over them, a great wind took it, and it was all blown away, and passed; and then a hush fell.
From what Gandalf has said of Sauron's destruction:
For he will lose the best part of the strength that was native to him in his beginning, and all that was made or begun with that power will crumble, and he will be maimed for ever, becoming a mere spirit of malice that gnaws itself in the shadows, but cannot again grow or take shape.
We can be almost certain that Saruman suffered the same fate.