Version:           2.0.2 (using KDE 3.5.10)
Compiler:          gcc-4.2.4 
OS:                Linux
Installed from:    Gentoo Packages

Like the title says, Kile drops its indentation when starting a new line because of static word wrap.  For example, normally I might have indentation like:

\begin{proof}
 One long line of the proof->
 wraps onto the next visual line.
\end{proof}

where -> means the line wrapped.  But with static word wrap, it comes out

\begin{proof}
 First static-wrapped line
Second static-wrapped line.
\end{proof}

The second line (and subsequent ones) lose the initial space.  This breaks the visual structure of the document.