Does there exist a function that is not a relation

No, a function is always a relation. A relation may not always a function but the vice versa is true.

It's like the relationship between a rectangle and a square. A rectangle has 4 right angles and 2 sets of congruent sides while a square has 4 right angles and all sides are congruent. Which means that a rectangle is a square, but a square may not always be a rectangle.