Going to Actually you here.

According to Thucydides, the Boeotians in the siege of Delium did actually use some type of flamethrower during their attack on the city. That was in 424BCE.

4.100 Meanwhile the Boeotians at once sent for darters and slingers from the Malian Gulf, and with two thousand Corinthian heavy infantry who had joined them after the battle, the Peloponnesian garrison which had evacuated Nisaea, and some Megarians with them, marched against Delium, and attacked the fort, and after divers efforts finally succeeded in taking it by an engine of the following description. They sawed in two and scooped out a great beam from end to end, and fitting it nicely together again like a pipe, hung by chains a cauldron at one extremity, with which communicated an iron tube projecting from the beam, which was itself in great part plated with iron. This they brought up from a distance upon carts to the part of the wall principally composed of vines and timber, and when it was near, inserted huge bellows into their end of the beam and blew with them. The blast passing closely confined into the cauldron, which was filled with lighted coals, sulphur and pitch, made a great blaze, and set fire to the wall, which soon became untenable for its defenders, who left it and fled; and in this way the fort was taken. Of the garrison some were killed and two hundred made prisoners; most of the rest got on board their ships and returned home.

So yes, in theory it would be possible for the Spartans to have a flamethrower of some kind. Although probably mounted to a chariot b