You Do Need power plug adapter while traveling from Liechtenstein to Chile.

Here is Why You Need An Adapter

The outlet and plug type used in Chile are C and L and in Liechtenstein are C and J. This is how they look like.

Liechtenstein

In Liechtenstein, outlets of type C and J are in use.

Type C

Type J

Chile

In Chile, outlets of type C and L are in use.

Type C

Type L