Well, I did it.
If it can be of any use to somebody else, I'll tell here what to do.
It is quite home-made and not optimized at all, but it works.
First, if you want the extra fields visibility to be configurable, add these lines in mod_fpss/mod_fpss.xml
(By default, visibility will be set to No)
<param name="extra_fields" type="list" default="" label="FPSS_EXTRA_FIELDS" description="FPSS_SHOW_OR_HIDE_THIS_ELEMENT_IN_THE_SLIDESHOW">
<option value="0">FPSS_NO</option>
<option value="1">FPSS_YES</option>
</param>
Add also the right string in languages/en-GB/en-GB.mod_fpss.ini
FPSS_EXTRA_FIELDS="Extra fields"
Then you'll have to retrieve these extra-fields from the database for each article from the category.
Complete the SQL request in com_fpss/helpers/slideshow.php, ~line 197
$query = "SELECT item.title, item.introtext, item.extra_fields, item.catid, item.alias,[...]
and add the extra-fields to the slide data, ~line 224
$slide->extra_fields = json_decode($row->extra_fields);
The "extra_fields" DB field is a text field that contains all the extra fields. It is encoded using JSON encoding.
The json_decode PHP function parses the field content into an array of extra fields headed "id" and "value".
Finally you may want to use the extra-fields stored in the slides data...
The following is an example built from what I wanted to do.
I wanted to print a specific extra-field as a subtitle on each slide. This extra-field had the id 36 in K2 > Extra Fields > "id" column
I thus added this piece of code in the slideshow template's default.php, between the slidetext divs, right under the title:
<?php if($slide->params->get('extra_fields')): ?> <- this only if a parameter was defined in mod_fpss.xml
<?php $extraFields = $slide->extra_fields; ?>
<?php foreach ($extraFields as $key=>$extraField):?>
<?php if ($extraField->id == '36'): ?><h1><?php echo $extraField->value; ?></h1><?php endif; ?>
<?php endforeach; ?>
<?php endif; ?>
It runs across all the extra-fields stored in the slide data to find the right id, then prints it once found.
Went lazy when trying to associate id and extra-field name, so I use it as it is and that's not so bad.