The strict pragma only protects the current namespace

No, remove the use strict; and the below will run without complaint.

package X;
use strict;
sub one { 1 }

package Y;
$yy = 1;

package main;
exit;

__DATA__
Global symbol "$yy" requires explicit package name at x line 6.
Execution of x aborted due to compilation errors.
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Be well,
rir

Update: added initial quote for clarity