Depends on what you mean by "like the Berner Oberland".
If you mean a scenic cul-de-sac alley, then perhaps the Klein Walsertal near Oberstdorf. Or Oberstdorf itself.
If you mean a high-altitude village, there's two options I can think of, although neither is auto-free. One, Obertauern near Salzburg. This is not any sort of traditional village, but a modern purpose-built winter ski resort built along a high Alpine road pass. The hiking is as good as anywhere else in the Alps. Only a few hotels remain open during the summer, but they are of a very high quality and offer significant off-season discounts for the level of service they provide. I also like this town because during heat waves, the high altitude provides significant relief.
Another high-altitude option is Flumserberg, in St. Gallen canton. This isn't one distinct village, but a series of hamlets spread across the side of the mountain. It functions as a sort of weekend getaway for Zurich residents- the international tourism industry seems to have completely overlooked the town. The Alps here aren't nearly as towering, but still very scenic. If you have mobility problems, one advantage of Flumserberg is that the town maintains several kilometers of nearly flat trails near the summit. Just a ski lift to the top.