D-Branes in $\textbf{AdS}_3\times \textbf{S}^3 \times \textbf{S}^3 \times \textbf{S}^1$

String theory on $\mathrm{ AdS}_3\times \mathrm{ S}^3 \times \mathrm{ S}^3 \times \mathrm{S}^1$ with two units of NS flux through each of the two $3$-spheres (and one unit of NS flux supporting the $\mathrm{AdS}_3$ factor) was recently argued to be exactly dual to the symmetric orbifold of eight free fermions and two bosons. This setup is interesting since it allows for a simple NS-R description. In this paper we study the spherical D-branes of this theory, and identify them with branes in the dual symmetric orbifold theory. We also comment briefly on the $\mathrm{ AdS}_2$ branes.