I have worked it out.
replace
#echo tep_draw_separator('pixel_trans.gif', '24', '15') . ' ' . tep_draw_input_field("extra_field[".$extra_fields['products_extra_fields_id']."]", $extra_product_field[$extra_fields['products_extra_fields_id']]);
with
echo tep_draw_textarea_field("extra_field[".$extra_fields['products_extra_fields_id']."]", 'soft', '70', '15', $extra_product_field[$extra_fields['products_extra_fields_id']]);
although you must have the same kind of input box for all of the custom fields. Also the database field is set to a character limit of 64.