We report on thermodynamic, magnetization, and muon spin relaxation measurements of the strong spin-orbit coupled iridate ${\mathrm{Ba}}_{3}{\mathrm{IrTi}}_{2}{\mathrm{O}}_{9}$, which constitutes a distinct frustration motif made up of a mixture of edge- and corner-sharing triangles. In spite of a strong antiferromagnetic exchange interaction of the order of 100 K, we find no hint for long-range magnetic order down to 23 mK. The magnetic specific heat data unveil $T$-linear and $T$-squared dependences at low temperatures below 1 K. At the respective temperatures, the zero-field muon spin relaxation features a persistent spin dynamics, indicative of unconventional low-energy excitations. A comparison to the $4d$ isostructural compound ${\mathrm{Ba}}_{3}{\mathrm{RuTi}}_{2}{\mathrm{O}}_{9}$ suggests that a concerted interplay of compasslike magnetic interactions and frustrated geometry promotes a dynamically fluctuating state in a triangle-based iridate.