there is a very subtle difference..it might also depend on your regional accent..."peek" is pronounced with a slight stretching of the 'e' sound, ie pe..ek, where as "peak" is pronounced without...
As another native speaker, no, there is no difference.
"peek" and "peak" are both pronounced identically \ˈpēk\
They are homophones.
Pick is pronounced \ˈpik\.
Piece and peace are also homophones and pronounced identically \ˈpēs\ while piss is pronounced \'pis\.
Apparently you have trouble distinguishing between ē and i sounds but there are only two different sounds in those word triplets, not three.
now try piece,peace and piss :)