Yes, it looks like they do. Within certain limits, though and with the limitation that only small pilot studies have been done.
A study assessing the the results of such a device in 15 men found that
After 6 months the mean gain in length was significant, meeting the goals of the effect size, at 2.3 and 1.7 cm for the flaccid and stretched penis, respectively. No significant changes in penile girth were detected.
A pilot phase-II prospective study to test the ‘efficacy’ and tolerability of a penile-extender device in the treatment of ‘short penis’
There is one meta study on the subject (a study looking at and summarizing other studies) that says
In conclusion, penile extenders appear to be an effective treatment for patients who complain of ‘short penis’. The application of such devices can be recommended in all patients regardless of the penile length, because of the low risk of complications
Non-invasive methods of penile lengthening: fact or fiction?
There's links to the original studies in that review.
Both studies are from the British Journal of Urology, which seems like a decently trustworthy journal and nothing in the papers stood out to me as unreasonable. The usual warnings apply, though: talk to you doctor about your plans and have one supervise the process if you both decide this is the best course of action for you.
So, yes, preliminary research seems like they have some effect (1 to 2 centimeters) if used as advised.