Never allow markdown cells to be editable.

jupyter_notebook_noneditable_markdown.js

// When a cell is clicked and the cell's mode is changed to the edit mode
// Then, this below event is called.
$([IPython.events]).on("edit_mode.Cell", function () {
    if (IPython.notebook.get_selected_cell().cell_type == "markdown") {
        IPython.notebook.execute_selected_cells();
    }
});