Most Markdown editors get the two-trailing-space rule wrong in their syntax highlighting, but get it right when actually exporting to HTML. Ulysses gets it wrong everywhere.
[EDIT: @UlyssesApp tweeted to explain that their app does manual line breaks with SHIFT+Enter, which exports as <br> in HTML and two trailing spaces in Markdown.]
The correct HTML result for the above snippet is:
<blockquote>
<p>This is a block quote<br />
that should<br />
continue to<br />
the next<br />
lines.</p>
</blockquote>
Ulysses generates the following incorrect HTML:
<blockquote>
<p>This is a block quote </p>
</blockquote>
<p>that should </p>
<p>continue to </p>
<p>the next </p>
<p>lines.</p>