  We report on thermodynamic, magnetization, and muon spin relaxation
measurements of the strong spin-orbit coupled iridate Ba$_3$IrTi$_2$O$_9$,
which constitutes a new frustration motif made up a mixture of edge- and
corner-sharing triangles. In spite of 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 the $T$-linear and
-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 Ba$_3$RuTi$_2$O$_9$ suggests that a concerted
interplay of compass-like magnetic interactions and frustrated geometry
promotes a dynamically fluctuating state in a triangle-based iridate.
