Colour can be found on many period leather items. There are lots of examples of extant artifacts in museums still showing the original colours, or at least enough traces that indicate they had been coloured originally. We also frequently see examples in artwork of objects that could be made of leather (frequently pouches) that are different colours.
There are really two types of examples, objects that were fully coloured, and objects that have designs painted on them (often in combination with leather tooling).
It's obviously difficult to tell from the artwork whether a coloured object is made of leather or fabric, but there are some ways to make an educated guess in some cases. For example, with the stick-purse the two extant examples with this design both have flaps made of leather (even the Mary Rose artifact where the pouches may have been made of fabric).
The focus of my A&S project is the period dyes, rather than the painted decorations, but I think it's important to understand the different uses of colour in period as part of that discussion.
Painting on Leather:
The Reverend's Big Blog of Leather has an interesting post on the history of leather painting (this is actually the first of a series of posts on the topic as he does some experiments on various techniques).
Another resource, this time on painted shields, was linked to in the previously mentioned blog post. Mostly about painted wood but has a few tidbits related to leather and a lot about period pigments.
Here's an excellent example of painting on leather. The case for the Imperial Crown of the Holy Roman Empire, 14th century The Kunsthistoriches Museum: http://bilddatenbank.khm.at/viewArtefact?id=100456
My badge carving project, while using modern paints/dyes, would be a similar example.
As mentioned in some previous posts, I have four different period dyers manuals that contain recipes for dying leather, which is pretty strong evidence that this was a practice done in period. Given that they are from several regions and time periods, it also suggests it was fairly widely done. Most of these recipes are for dying the whole skin as part of the preparation of the leather. The coloured skin would then be used for making leather items (like pouches, clothing etc.)
To show that these dyes can work, I point to Karl Robinson, a leatherworker and merchant in the UK. He uses period dyes from one of my four sources for his leather items.
I believe there are references in Goubitz's Purses in Pieces of coloured leather, I'll have scan it for mentions of colouring as part of my documentation.
Since the trigger for this project was my stick-purse, I've got some examples from that research that apply here. There are several period paintings of stick-purses with coloured flaps. Based on the extant examples, those flaps are probably made from leather and it's likely the leather for these flaps would have been dyed rather than painted. Here's one artwork example.
The Banker and His Wife, REYMERSWAELE, Marinus van (16th century)