You Do Need power plug adapter while traveling from Saint Barthélemy to Gibraltar.

Here is Why You Need An Adapter

The outlet and plug type used in Gibraltar is G and in Saint Barthélemy are C and E. This is how they look like.

Saint Barthélemy

In Saint Barthélemy, outlets of type C and E are in use.

Type C

Type E

Gibraltar

In Gibraltar, outlets of type G are in use.

Type G