Fire crews were called to the blaze at the building, which houses a walk-in centre that is only open during the daytime, at 22:59 BST on Tuesday.
Images shared on social media show the roof of the three-storey building engulfed in flames. Police tweeted to ask people to avoid the area around the High Street.
No-one is believed to have been injured in the blaze.
It is feared there could be flammable oxygen cylinders at the site. The fire service expects to be at the scene for several more hours.
A large fire has ripped through Weybridge Community Hospital in Surrey.