We study mimetic gravity in the presence of a Dirac–Born–Infeld (DBI)-like term, which is a non-canonical setup of the scalar field’s derivatives. We consider two general cases with varying and constant sound speeds and construct the potentials for both the DBI and mimetic DBI (MDBI) models. By considering the power-law scale factor as a=a0t n, we search for the observational viability of these models. We show that the MDBI model in some ranges of the parameter space is free from ghost and gradient instabilities. By studying the behavior of r–ns and αs–ns in confrontation with Planck2018 data, we find some constraints on the model’s parameters. We show that, for the case with varying sound speed, power-law DBI inflation is not consistent with Planck2018 TT, TE, EE+lowE+lensing data, but the MDBI inflation is consistent with the same data at 95% confidence level, in some ranges of the model’s parameter space such as 40n55; the model is also free from instabilities in these ranges of parameters. For a constant sound speed, we study both DBI and MDBI models numerically by adopting some sample values of cs, and find n∼102 for the DBI model and n∼10 for the MDBI model. We also compare the results with Planck2018 TT, TE, EE+lowE+lensing +BK14+BAO data and see that the DBI and MDBI models with varying sound speed are ruled out by these joint data. However, these models with constant sound speed are consistent with the same data, with n∼102 for the DBI model and n∼10 for the MDBI model. In this case, we find some tighter constraints on the corresponding sound speed.
