Just for reference, I used a simple query to get a response from the root server, which is presented with XML 1.0 declaration but contains the invalid unicode character in the results. Then I changed the header to 1.1 and tried running it through several online validators that claim to have 1.1 support, but all rejected the character.
The document I ended up with is
<?xml version="1.1" encoding="UTF-8"?>
<BESAPI xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="BESAPI.xsd">
<Query Resource="character 20">
(the unprintable character is here in the post but is not printable/displayable. But this doc includes the query so you could reproduce pretty easily.
If you send this through a browser to api/query, the browser complains the result page is invalid XML but I could “View Source” to get the XML response)