  We study mimetic gravity in the presence of a 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 models. By considering the power-law scale factor
as $a=a_{0}\,t^{n}$, we seek for the observational viability of these models.
We show that, the Mimetic DBI model in some ranges of the parameters space is
free of ghost and gradient instabilities. By studying $r-n_{s}$ and
$\alpha_{s}-n_{s}$ behavior in confrontation with Planck2018 data, we find some
constraints on the model's parameters. We show that for the case with varying
sound speed, although power-law DBI inflation is not consistent with Planck2018
TT, TE, EE+low E+lensing data, but the Mimetic DBI inflation is consistent with
Planck2018 TT, TE, EE+low E+lensing data at 95$\%$ CL, in some ranges of the
model's parameters space as $40\leq n \leq 55$ where the model is
instabilities-free in these ranges of parameters too. For the constant sound
speed, by adopting some sample values of $c_{s}$, we study both DBI and Mimetic
DBI model numerically and find $n\sim 10^{2}$ for DBI model and $n\sim 10$ for
Mimetic DBI model. We also compare the results with Planck2018 TT, TE, EE+low
E+lensing+BK14+BAO data and see that the DBI and Mimetic DBI model with varying
sound speed are ruled out with these joint data. However, these models with
constant sound speed are consistent with Planck2018 TT, TE, EE+low
E+lensing+BK14+BAO data with $n\sim 10^{2}$ for DBI model and $n\sim 10$ for
Mimetic DBI model. In this case, we find some tighter constraints on the
corresponding sound speed.
