No.

The following works:

my $DEBUG;
BEGIN { $DEBUG = $ARGV[0]; }
use if $DEBUG, 'strict';
# <-- The compiler is here because
#     Perl executes use statements
#     as soon as it compiles them.
#     "strict" will work until the
#     end of this block (namely
#     until the end of the file).
print($test);   # strict error if $DEBUG
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The following works:

my $DEBUG;
BEGIN {
   $DEBUG = $ARGV[0];
   if ($DEBUG) {
      require strict;
      import strict;
   }
}
# <-- The compiler is here because
#     Perl executes BEGIN blocks
#     as soon as it compiles them.
#     "strict" will work until the
#     end of this block (namely
#     until the end of the file).
print($test);   # strict error if $DEBUG
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The point, however, is that the following doesn't work:

my $DEBUG;
BEGIN {
   $DEBUG = $ARGV[0];
   if ($DEBUG) {
      eval <<'__EOI__';
         use strict;
         # <-- The compiler instance used
         #     by eval is here because
         #     Perl executes use statements
         #     as soon as it compiles them.
         #     "strict" will work until the
         #     end of this block (namely
         #     until the end of the code
         #     passed to eval).
__EOI__
   }
}
print($test);   # No strict error
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Actually, I thought the second one wouldn't work either, but it does for the reason I gave in the comments.