I had to unslick the carousel before the ajax call starts, but you can't do that until there is already a slick carousel. So, I set a variable to 0 and only run unslick after it changed
var slide = 0
if(slide>0)
$('#ui-id-1').slick('unslick');
$.ajax({
//do stuff, here
},
success: function( data ) {
$('#ui-id-1').slick();
slide++;
}