Can creatures attacked with Burn set other creatures on fire?

Small Fire Elementals (amongst others) have an attack called Burn:

A fire elemental deals fire damage in addition to damage dealt on a successful hit in melee. Those affected by the burn ability must also succeed on a Reflex save or catch fire, taking the listed damage for an additional 1d4 rounds at the start of its turn. A burning creature can attempt a new save as a full-round action. Dropping and rolling on the ground grants a +4 bonus on this save. Creatures that hit a burning creature with natural weapons or unarmed attacks take fire damage as though hit by the burning creature and must make a Reflex save to avoid catching on fire. See Burn for more details.

Since creatures that hit a burning creature take fire damage and must make a save, does this mean that if an elemental successfully Burns an elf, and then that elf is punched by a goblin, the goblin takes fire damage and might be set on fire?