extending the example to more than 2 frames in beamer

Question

I have an example which is 3 slides long, as shown in the figure. I would like to have all 3 boxes green and have the same number This is my code, basically i changed everywhere to "example"

\documentclass{beamer}
\usetheme{Warsaw}
\usepackage{lipsum}

\makeatletter
\newenvironment<>{examples}[1][\proofname]{%
    \par
    \def\insertexamplename{#1\@addpunct{.}}%
    \usebeamertemplate{example begin}#2}
  {\usebeamertemplate{example end}}
\makeatother

\begin{document}

\begin{frame}
\begin{examples}
\lipsum[1]
\end{examples}
\end{frame}

\begin{frame}
\begin{examples}[\proofname\ (Cont.)]
\lipsum[1]
\end{examples}
\end{frame}

\begin{frame}
\begin{example}[\examplename\ (Cont.)]
\lipsum[1]
\end{example}
\end{frame}

Answer 1

If I get your question right (I hope so) the following could work. It uses tcolorbox and not the default proof-boxes from beamer. I don't know how to do this with beamer. For that, you might have a look at the beamer documentation.

\documentclass{beamer}
\usetheme{Warsaw}
\usepackage{lipsum}
\usepackage[skins]{tcolorbox}

\definecolor{proofhead}{HTML}{006000}
\definecolor{proofbody}{HTML}{e6efe6}
\newcounter{proofs}
\newtcolorbox{myproofbox}[1][]{%
    beamer,%
    colback=proofbody,%
    colframe=proofhead,%
    boxsep=0pt,
    top=3pt,
    bottom=5pt,
    left=4pt,%
    right=4pt,%
    fuzzy shadow={1mm}{-1mm}{0mm}{0.1mm}{black!50!white},%
    #1%
}

\newenvironment<>{myproof}[1][]{%
    \refstepcounter{proofs}
    \begin{myproofbox}[%
        adjusted title=\normalfont{\proofname~\theproofs~#1},% you might change this line to set a default title
    ]}
    {\end{myproofbox}}

\begin{document}
    \begin{frame}
        This is how it should look like:
        \begin{example}
            \lipsum[2]
        \end{example}
    \end{frame}
    \begin{frame}
        This is how my code looks like:
        \begin{myproof}
            \lipsum[2]
        \end{myproof}
    \end{frame}
    \begin{frame}
        \addtocounter{proofs}{-1}% This reduces the counter so it's the same number
        \begin{myproof}[(Cont.)]
            \lipsum[2]
        \end{myproof}
    \end{frame}
\end{document}

To set a complete custom title you can also use the following instead of \begin{myproof}...\end{myproof}:

\begin{myproofbox}[adjusted title=\normalfont{<Insert your Title here>}]
    <Insert the content of the box here>
\end{myproofbox}

Note that this doesn't step the counter. To step the counter in a way that you can label the box you should use \refstepcounter{proofs}. To use the counter value (e.g. in your title) you can use \theproofs or something like \roman{proofs} for other formatting of the number.