What's wrong? It doesn't work, but it can be fixed.
__FILE__ is the special literal that I need, not __FILENAME__. This works:

#! perl -w
use strict;

package This::Module;
our $_Filename = __FILE__;

#        ...

package main;
print "$This::Module::_Filename\n";
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I found this in perldata (which I should have read beforehand)
The special literals __FILE__, __LINE__, and __PACKAGE__ represent the current filename, line number, and package name at that point in your program.

However, after I replaced __FILENAME__ by __FILE__ in your code, I had trouble getting the constant out of the package.

#! perl -w
use strict;

package This::Module;
use CONSTANT _Filename => __FILE__;

#        ...

package main;
#print "$This::Module::_Filename\n";
#Use of uninitialized value in concatenation (.) at getfilename1.pl li
+ne 10.

print " @{[ This::Module::_Filename ]}\n";
#Bareword "This::Module::_Filename" not allowed while "strict subs" in
+ use at getfilename1.pl line 13.
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Now I am a little lost. Questions:
What is the syntax for getting a package constant outside the package? Should I export it?
Is 'use CONSTANT' same as 'use constant', from constant.pm?
TIA
Rudif