While I am unfamiliar with Cache::Mmap, your error suggests that it is using Storable internally. You are trying to store a simple value, but Storable only works with references. Try changing your code like so: (untested)

my $key1 = "Test";
my $val1 = $cache->read($key1);
print "read key1 count: ",$val1->{count},"\n";
$val1->{count}++;
print "val1 count:",$val1->{count},"\n";
$cache->write($key1,$val1);
$val1 = $cache->read($key1);
print "read key1 count after write: ",$val1->{count},"\n";
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Note the change: the values stored in the cache are now hashrefs. You could also use scalar references, reading them with $$val1 and incrementing with ${$val1}++.

Lastly, your code as written will not compile under strict. While it would not have caught this issue, good practice is to always use strict; and use warnings;.