Can sodium and chlorine form an ionic compound?

YES!!! THis is the only form of combination between the two elements.

There is an experiment demonstrating this reaction.

In a closed chlorine containing gas jar, insert a small pellet of sodium. There is an immediate reaction as the two elements react together to form sodium chloride.

2Na(s) + Cl2(g) = 2NaCl(s)

Sodium chloride is common table salt, that you use in cooking and tasting food.