Hello & welcome, I am Dr. B, a licensed veterinarian and I would like to help you with your wee one today.
Now we always need to be careful when it comes to possible constipation in the hamster. The reason is because often we see hamsters misdiagnosed with constipation. For example, people often think bloated hamsters are constipated when actually they may have fluid accumulation in their abdomen or masses (and where slow fecal passage is due to compression of the GI by the distending agent). As well, often people assume material attached to the rectum is "stuck feces," where it could also be a prolapsed bit of rectum sticking out that is discoloring as the tissue dies.
In this case, this discolored brown lump at the rectum does have me concerned for Harriet. I appreciate that you don't want to stress her but it is important to know what this is since it will dictate how serious this issue is and potentially determine her prognosis. Therefore, you may want to consider trying to sponge off this brown material if it will come away or carefully bathe her backside. Do not pull the material away from her rectum. If you clear away feces and can see a protrusion of gut tissue (the tissue tends to be red/pink to start but can dull as it gets congested from being trapped outside the rectum), then we'd be concerned about rectal prolapse. If this is present, it is an emergency situation that needs to be addressed immediately (since rectal tissue is not meant to be outside the body and if not replaced the tissue can die and thus cost the hamster its life).
Otherwise, if this is fecal material protruding from the rectum, then we can consider constipation being the actual issue here. The warm water can help her but otherwise oral administration of GI lubricants can be useful. Olive oil isn't typically very helpful (since it gets digested part want through the GI if they will ingest it) in these cases. Instead, you may consider treating her with a small volume of cat hairball medication (ie. Catalax) to lubricate the gut and can facilitate the movement of hard feces out of the GI. . Alternatively, you can administer a small volume of Miralax, lactulose (LINK) or mineral oil orally (ideally offer in food or if given via oral syringe then take care to avoid aspiration, since that would cause problems we'd best avoid). While she won't be keen to take the liquids by mouth on their own, you can mix them with veggie baby food or fruit juice to get her to take it for you and get things moving for her.
So, I appreciate that you don't want to stress Harriet but if she is in severe discomfort and we cannot be sure a prolapse isn't present here, I would advise some gentle handling and cleaning would be prudent. Depending on your findings, you will need to take one of the above approaches for her.
Finally, just since you noted that your vet's are closed for the evening, I do want to note if you want to have her seen now, most veterinary practices here do have contingency plans for emergency care for their patients (since they are legally obligated to provide an out of hours service). This means that if you ring the practice, they will likely have a message to direct you on how to contact their emergency service. And if you don't have a vet you can find one local to you, you can check you can check the RCVS Register (HERE) to find your local Vets Now (LINK) who are open all nights/weekends.
I hope this information is helpful.
If you need any additional information, do not hesitate to ask!
All the best,
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