For the input you gave, it won't make a difference, but if $Y is a constant string, not a regexp, you should escape it, as follows:

foreach $Y (@dependantFiles)
{
    foreach $X (@namedFiles)
    {        
        if(  $X =~ m/\Q$Y\E/i)
        {
            print "Found it\n"
            last;
        }
    }
}
[download]

I also removed the useless 'g' option and the useless 'next'.

But if it's not a regexp, why use m// at all! index is much faster:

foreach $Y (@dependantFiles)
{
    foreach $X (@namedFiles)
    {        
        if (index(lc($X), lc($Y)) >= 0)
        {
            print "Found it\n"
            last;
        }    
    }
}
[download]

With a slight optimization:

foreach $Y (@dependantFiles)
{
    my $lcY = lc($Y);

    foreach $X (@namedFiles)
    {        
        if (index(lc($X), $lcY) >= 0)
        {
            print "Found it\n"
            last;
        }    
    }
}
[download]

If the $Y will always be at the end of $X, the following is even faster and more accurate:

foreach $Y (@dependantFiles)
{
    my $lcY = lc($Y);

    foreach $X (@namedFiles)
    {        
        if (lc(substr($X, -length($Y))) eq $lcY)
        {
            print "Found it\n"
            last;
        }    
    }
}
[download]

I haven't solved the problem where $X is c:\path\ab.exe and $Y is b.exe. There are a couple of solutions, but I'll let you try to solve it before I say more.