Yes,Botanical Gardens are living herbaria as herbaria means keeping specimens in an herbarium sheets,as when we do so we actually need to kill the specimen but in botanical gardens they are preserved alive and a appropriate atmosphere(required to live) are maintained.
Herbarium is the collection of preserved plants Specimen. The specimen preserved may be, any part of the plant or sometimes the whole plant. Herbarium is formed to preserve the plant species, but in nonliving form. In the same way a botanical garden also preserve the plants. But in their natural and living condition. So they can be called as, living herberia.
It is a collection of pressed, dried and preserved plants mounted on paper sheets, properly labeled and systematically arranged. It is for reference or study. Maintained mostly by research laboratories, colleges, botanical garden and museums.A book containing an orderly collection of dried plants is also called a herbarium.