The production yield and angular anisotropy of prompt $\mathrm{D_s^+}$ mesons were measured as a function of transverse momentum ($p_\mathrm{T}$) in Pb-Pb collisions at a centre-of-mass energy per nucleon pair $\sqrt{s_\mathrm{NN}}=5.02~\mathrm{TeV}$ collected with the ALICE detector at the LHC. $\mathrm{D_s^+}$ mesons and their charge conjugates were reconstructed at midrapidity ($|y|<0.5$) from their hadronic decay channel $\mathrm{D_s^+\to \phi\pi^+}$, with $\mathrm{\phi\to K^-K^+}$, in the transverse momentum intervals $2<p_\mathrm{T}<50~\mathrm{GeV}/c$ and $2<p_\mathrm{T}<36~\mathrm{GeV}/c$ for the 0-10% and 30-50% centrality intervals, respectively. For $p_\mathrm{T}>10~\mathrm{GeV}/c$, the measured $\mathrm{D_s^+}$-meson nuclear modification factor $R_\mathrm{AA}$ is consistent with the one of non-strange D mesons within uncertainties, while at lower $p_\mathrm{T}$ a hint for a $\mathrm{D_s^+}$-meson $R_\mathrm{AA}$ larger than that of non-strange D mesons is seen. The enhanced production of $\mathrm{D_s^+}$ relative to non-strange D mesons is also studied by comparing the $p_\mathrm{T}$-dependent $\mathrm{D_s^+}/\mathrm{D^0}$ production yield ratios in Pb-Pb and in pp collisions. The $\mathrm{D_s^+}/\mathrm{D^0}$ ratio measured in Pb-Pb collisions is found to be on average higher than that in pp collisions in the transverse-momentum interval $2<p_\mathrm{T}<8~\mathrm{GeV}/c$ with a significance of $2.3\sigma$ and $2.4\sigma$ for the 0-10% and 30-50% centrality intervals, respectively. The azimuthal anisotropy coefficient $v_2$ of prompt $\mathrm{D_s^+}$ mesons was measured in Pb-Pb collisions in the 30-50% centrality interval and it is found to be compatible with that of non-strange D mesons. The main features of the measured $R_\mathrm{AA}$, $\mathrm{D_s^+}/\mathrm{D^0}$ ratio, and $v_2$ as a function of $p_\mathrm{T}$ are described by theoretical calculations of charm-quark transport in a hydrodynamically expanding quark-gluon plasma including hadronisation via charm-quark recombination with light quarks from the medium. The $p_\mathrm{T}$-integrated production yield of $\mathrm{D_s^+}$ mesons is compatible with the prediction of the statistical hadronisation model.