You could use the `maparg()` function.

To test if the user mapped something to `<C-c>` in normal mode, you would write:

    if !empty(maparg('<C-c>', 'n'))

If the user mapped something, to store the `{rhs}` in a variable, you would write:

    let rhs_save = maparg('<C-c>', 'n')

If you want more information about the mapping, like:

 - is it silent (`<silent>` argument)?
 - is it local to the current buffer (`<buffer>` argument)?
 - is the `{rhs}` the evaluation of an expression (`<expr>` argument)?
 - does it remap the `{rhs}` (`nnoremap` vs `nmap`)?
 - if the user has another mapping which begins with `<C-c>`, does Vim wait for more characters to be typed (`<nowait>` argument)?
 - ...

Then, you could give a third and a fourth argument: `0` and `1`.  
`0` because you're looking for a mapping and not an abbreviation, and `1` because you want a dictionary with a maximum of information and not just the `{rhs}` value:

    let map_save = maparg('<C-c>', 'n', 0, 1)

----------

Assuming the user didn't use any special argument in his mapping, and that it doesn't remap the `{rhs}`, to restore it, you could simply write:

    let rhs_save = maparg('<C-c>', 'n')

    " do some stuff which changes the mapping

	exe 'nnoremap <C-c> ' . rhs_save

Or to be sure and restore all the possible arguments:

    let map_save = maparg('<C-c>', 'n', 0, 1)

    " do some stuff which changes the mapping

    exe (map_save.noremap ? 'nnoremap' : 'nmap') .
         \ (map_save.buffer ? ' <buffer> ' : '') .
         \ (map_save.expr ? ' <expr> ' : '') .
         \ (map_save.nowait ? ' <nowait> ' : '') .
         \ (map_save.silent ? ' <silent> ' : '') .
         \ ' <C-c> ' .
         \ map_save.rhs

----------

Edit: Sorry, I've just realised it wouldn't work as expected if the user calls a script-local function in the `{rhs}` of the mapping.

Suppose that the user has the following mapping inside his `vimrc`:

    nnoremap <C-c> :<C-U>call <SID>FuncA()<CR>
    
    function! s:FuncA()
        echo 'hello world!'
    endfunction

When he hits `<C-c>`, it displays the message `hello world!`.

And in your plugin, you save a dictionary with all the information, then temporarily change his mapping like this:

    let map_save = maparg('<C-c>', 'n', 0, 1)
    nnoremap <C-c> :<C-U>call <SID>FuncB()<CR>
    
    function! s:FuncB()
        echo 'bye all!'
    endfunction

Now, it will display `bye all!`.
Your plugin does some work, and when it's over, it tries to restore the mapping with the previous command.

It will probably fail with a message looking like this:

    E117: Unknown function: <SNR>61_FuncA

`61` is just the identifier of the script in which your mapping command would be executed. It could be any other number. If your plugin is the 42th file sourced on the user's system, it will be `42`.

Inside a script, when a mapping command is executed, Vim automatically translates the notation `<SID>` into the special key code `<SNR>`, followed by a number which is unique for the script, and an underscore. It has to do this, because when the user will hit `<C-c>`, the mapping will be executed outside of the script, and thus it won't know in which script `FuncA()` is defined.

The problem is that the original mapping was sourced in a different script than your plugin, so here the automatic translation is wrong. It uses the identifier of your script, while it should use the identifier of the user's `vimrc`.

But you could do the translation manually. The dictionary `map_save` contains a key called `'sid'` whose value is the correct identifier.  
So, to make the previous restoration command more robust, you could replace `map_save.rhs` with:

    substitute(map_save.rhs, '<SID>', '<SNR>' . map_save.sid . '_', 'g')

If the `{rhs}` of the original mapping contained `<SID>`, it should be properly translated. Otherwise, nothing should be changed.

And if you want to shorten the code a little, you could replace the 4 lines which take care of the special arguments with:

    join(map(['buffer', 'expr', 'nowait', 'silent'], 'map_save[v:val] ? "<" . v:val . ">": ""'))

The `map()` function should convert each item from the list `['buffer', 'expr', 'nowait', 'silent']` into the corresponding mapping argument but only if its key inside `map_save` is non-zero. And`join()` should join all the items into a string.

So, a more robust way of saving and restoring the mapping could be:

    let map_save = maparg('<C-c>', 'n', 0, 1)

    " do some stuff which changes the mapping

    exe (map_save.noremap ? 'nnoremap' : 'nmap') .
        \ join(map(['buffer', 'expr', 'nowait', 'silent'], 'map_save[v:val] ? "<" . v:val . ">": ""')) .
        \ map_save.lhs . ' ' .
        \ substitute(map_save.rhs, '<SID>', '<SNR>' . map_save.sid . '_', 'g')

----------

Edit2:

I'm facing the same issue as you, how to save and restore a mapping in a drawing plugin. And I think I found 2 issues that the initial answer didn't see at the time I wrote it, sorry about that.

First issue, suppose that the user uses `<C-c>` in a global mapping but also in a buffer-local mapping. Example:

    nnoremap          <C-c>    :echo 'global mapping'<CR>
    nnoremap <buffer> <C-c>    :echo 'local  mapping'<CR>

In this case, `maparg()` will give priority to the local mapping:

    :echo maparg('<C-c>', 'n', 0, 1)

    ---> {'silent': 0, 'noremap': 1, 'lhs': '<C-C>', 'mode': 'n', 'nowait': 0, 'expr': 0, 'sid': 7, 'rhs': ':echo ''local  mapping''<CR>', 'buffer': 1}

Which is confirmed in `:h maparg()`:

		The mappings local to the current buffer are checked first,
		then the global mappings.

But maybe you aren't interested in the buffer-local mapping, maybe you want the global one.  
The only way I found to, reliably, get the information about the global mapping, is to try to temporarily unmap a potential, shadowing, buffer-local mapping using the same key.

It could be done in 4 steps:

 1. save a (potential) buffer-local mapping using the key `<C-c>`
 2. execute `:silent! nunmap <buffer> <C-c>` to delete a (potential) buffer-local mapping
 3. save the global mapping (`maparg('<C-c>', 'n', 0, 1)`)
 4. restore the buffer-local mapping

The second issue is the following. Suppose that the user didn't map anything to `<C-c>`, then the output of `maparg()` will be an empty dictionary. And in this case, the restoration process doesn't consist in the installation of a mapping (`:nnoremap`), but in the destruction of a mapping (`:nunmap`).

To try to solve these 2 new issues, you could try this function to save mappings:

    fu! Save_mappings(keys, mode, global) abort
        let mappings = {}
    
        if a:global
            for l:key in a:keys
                let buf_local_map = maparg(l:key, a:mode, 0, 1)
    
                sil! exe a:mode.'unmap <buffer> '.l:key
    
                let map_info        = maparg(l:key, a:mode, 0, 1)
                let mappings[l:key] = !empty(map_info)
                                    \     ? map_info
                                    \     : {
                                            \ 'unmapped' : 1,
                                            \ 'buffer'   : 0,
                                            \ 'lhs'      : l:key,
                                            \ 'mode'     : a:mode,
                                            \ }
    
                call Restore_mappings({l:key : buf_local_map})
            endfor
    
        else
            for l:key in a:keys
                let map_info        = maparg(l:key, a:mode, 0, 1)
                let mappings[l:key] = !empty(map_info)
                                    \     ? map_info
                                    \     : {
                                            \ 'unmapped' : 1,
                                            \ 'buffer'   : 1,
                                            \ 'lhs'      : l:key,
                                            \ 'mode'     : a:mode,
                                            \ }
            endfor
        endif
    
        return mappings
    endfu

... and this one to restore them:

    fu! Restore_mappings(mappings) abort
    
        for mapping in values(a:mappings)
            if !has_key(mapping, 'unmapped') && !empty(mapping)
                exe     mapping.mode
                   \ . (mapping.noremap ? 'noremap   ' : 'map ')
                   \ . (mapping.buffer  ? ' <buffer> ' : '')
                   \ . (mapping.expr    ? ' <expr>   ' : '')
                   \ . (mapping.nowait  ? ' <nowait> ' : '')
                   \ . (mapping.silent  ? ' <silent> ' : '')
                   \ .  mapping.lhs
                   \ . ' '
                   \ . substitute(mapping.rhs, '<SID>', '<SNR>'.mapping.sid.'_', 'g')
    
            elseif has_key(mapping, 'unmapped')
                sil! exe mapping.mode.'unmap '
                                    \ .(mapping.buffer ? ' <buffer> ' : '')
                                    \ . mapping.lhs
            endif
        endfor
    
    endfu

----------

The `Save_mappings()` function could be used to save mappings.  
It expects 3 arguments:

 1. a list of keys; example: `['<C-a>', '<C-b>', '<C-c>']`
 2. a mode; example: `n` for normal mode or `x` for visual mode
 3. a boolean flag; if it's `1`, it means you're interested in global mappings, and if it's `0`, in local ones

With it, you could save the global mappings using the keys `C-a`, `C-b` and `C-c`, in normal mode, inside a dictionary:

    let your_saved_mappings = Save_mappings(['<C-a>', '<C-b>', '<C-c>'], 'n', 1)

Then, later, when you'll want to restore the mappings, you could call `Restore_mappings()`, passing the dictionary containing all the info as an argument:

    call Restore_mappings(your_saved_mappings)