Heat can only be generated at the tip of the carbon brush if there is an electrical fault on one of the armature windings or if the brush hasn't been sitting firmly on the commutator. The brush could have stuck in the brush holder and as the brush wears away it will leave a gap between itself and the commutator. Current will arch across that gap and heat will be generated - a similar situation to welding.
A photo of the other brush and commutator may have also helped, but you weren't to know that.
You may also find a slightly raised commutator segment if you look hard enough which will indicate a winding fault. In any case, it will need a new armature, brushes, springs and rear brush holder/end cap to get it working again.
As I doubt they carry those kind of spares, it will be a replacement pump.