Facet API select
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FacetAPI select
This module provides a facetapi widget to display a facet as a single-valued select box. It defaults to hiding the form submit button and submitting automatically but this can be turned off and degrades without javascript.
It was originally a patch to Jody Lynn's sandbox of the same name but it turned into more of a rewrite so a seperate repo was helpful. The behaviour is different as described in #1412442: Selected value.
I also just noticed a new module FacetAPI multiselect that also falls in this realm.
Usage
This module relies on a patch to the FacetAPI module (#1393928: Add a per-facet setting that allows only one item to be active at a time) which enables sane behaviour for a single-valued checkbox.
After applying that patch, select this widget for your given facet. In the global settings on the "Configure facet display" page set your operator to "OR" and check the box labelled "Limit to one active item".