You need to quotemeta the string in $p:

$p = $s = "^xxx";
print "$p has $s\n" if $s =~ /\Q$p/; #does print
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(Depending on whether there's something else not to be quoted after the $p (in your real case), you might also want \E to end the quoting, e.g. /\Q$p\E|foo/)

The \Q solution makes the most sense in my case because I dont need to check if there happens to be a caret in front.

Much appreciated.

print "$p has $s\n" if 0 == index $s, $p;
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$p = $s = "\^xxx";
print "$p has $s\n" if $s =~ /$p/; #does not print
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$p = $s = "\^xxx";

You probably meant

$s = "^xxx";
$p = '\^xxx';  # or "\\^xxx"
...
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(a backslash in a double-quoted string just escapes the next char...)

Why worry about quoting constructs? qr// has worked very well for ages:

my $p = qr/\^xxx/;