We consider stochastic 2D-Stokes equations with unbounded delay in fractional power spaces and moments of order p ≥ 2 driven by a tempered fractional Brownian motion (TFBM) B σ , λ (t) with − 1 / 2 < σ < 0 and λ > 0. First, the global existence and uniqueness of mild solutions are established by using a new technical lemma for stochastic integrals with respect to TFBM in the sense of p th moment. Moreover, based on the relations between the stochastic integrals with respect to TFBM and fractional Brownian motion, we show the continuity of mild solutions in the case of λ → 0 , σ ∈ (− 1 / 2 , 0) or λ > 0 , σ → σ 0 ∈ (− 1 / 2 , 0). In particular, we obtain p th moment Hölder regularity in time and p th polynomial stability of mild solutions. This paper can be regarded as a first step to study the challenging model: stochastic 2D-Navier–Stokes equations with unbounded delay driven by tempered fractional Gaussian noise. [ABSTRACT FROM AUTHOR]