  We construct new solutions of ten-dimensional supergravity with sixteen
supercharges which describe the backreaction of D$p$-branes with spherical
worldvolume. These solutions are holographically dual to the
$(p+1)$-dimensional maximally supersymmetric Yang-Mills theory on $S^{p+1}$.
The finite size of the sphere provides an IR cut-off for the gauge theory which
is manifested in the supergravity solution as a smooth cap-off of the geometry.
In the UV the size of the sphere plays no role and the backgrounds asymptote to
the well-known supergravity solutions that describe the near-horizon limit of
flat D$p$-branes. We compute the on-shell action of our spherical brane
solutions and show that it is in agreement with recent supersymmetric
localization results for the free energy of maximal SYM theory on $S^{p+1}$.
