wxTE_MULTILINE actually is a function which has some value, and that value seems to be not the string "wxTE_MULTILINE". So you can do it like this:

...

my $style = wxTE_MULTILINE();

Wx::TextCtrl->new( $self, -1, "", [5, 10], [-1, -1], $style );
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Update: Fixed a typo spotted by Fletch