Teachers and pupils at Meoncross School in Stubbington, Hants, were amazed when they saw the creature through the window during a lesson.
Since the squirrel, now nicknamed Pete, was first seen, it has become a regular fixture at the school but no one has been able to say whether the animal has fallen into purple paint, had a run-in with some purple dye, or whether there is another explanation.
Dr Mike Edwards, an English teacher, said: "I was sitting in my classroom and looked out the window and saw it sitting on the fence. I had to do a double take.
"Since then it's been a bit of a regular at the school - everyone's seen it.
"We thought it might have been paint or something but then when you look at it up close, it's an all over coat, not in patches like you'd expect if it had been near some paint.
"Its fur actually looks purple all the way through. It's an absolute mystery."
Pupils, staff and parents have contacted vets and even e-mailed television nature expert Bill Oddie to see if an explanation could be found.
Lorraine Orridge, the school's registrar, believes Pete's coloured fur looks like a school uniform.
She said: "The squirrel has become a bit of a legend among staff and pupils at the school.
"He makes an appearance most days and we always look forward to seeing him.
"We don't think he is a mutant squirrel but he may have had a mishap around the school.
"The old building where we have seen him nipping in and out is a bit of a graveyard for computer printers. He may have found some printer toners in there.
"We haven't seen any purple baby squirrels yet."
TV wildlife expert Chris Packham believes Pete will moult and lose his purple fur in time for spring.
He said: "I have never seen anything like it before.
"Squirrels will chew anything even if it's obviously inedible. It is possible he has been chewing on a purple ink cartridge and then groomed that colouring into his fur.
"Alternatively he may have fallen into a bucket containing a weak colour solution that has stained his fur.
"Underneath there's a normal grey squirrel who has just given himself an unusual hair colour - you would pay a fortune for that in some salons."
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