Is Black mulberry always deciduous?
WARNING: This answer is very tangential to the question. Confused Morus sp. with Rubus frutticosa (don't use "common name in questions"! luckily I wrote the answer using scientific names). So, as addendum: some trees tend to lose some leaves later (in my region, such trees can recover easily with broken branches).

"Always" is "always" a strong word in biology.

You describe your mulberry (Rubus sp.) as deciduous: the leaves are changed every year.

Many plants (especially trees) lose leaves in autumn/winter and they get new one in late winter/spring, few plants are marcescent, e.g. various oaks (Quercus sp.), so the leaves dry in winter and the plant lose it just when the new leaves appeaers. Some plants lose the leaves in summer.

Mulberries, like various "small plants", tend to keep the leaves until they form the new one (eventually, replacing all the shot). In my zone mulberries tend to keep old leaves until the new one are already developing.

Why this habit? Mulberries lives at borders of forests, so they try to have more energy/sun when the tallest tree lose leaves. On the other hand, being shrub and tending to radicate easily, it keeps safer in case of frosts, and it is also not a long living organism (as "plant", not as clone).

Note some decidouos young trees have also the same habits: they keep their few leaves in winter, to try to catch up. Being young also means few investment for the original plant, so it could be still a gaining habit. Look this winter on undergrowth.

Not so different to strawberries and many grasses, which replace the leaves in springs. But they are not "decidouos", just because they doesn't have a trunk or just they replace the entire shot the shots.