I believe the answer is yes and follows from the combination of Theorem 4.6 here https://arxiv.org/pdf/math/0111245.pdf and Theorem 1.3 here https://arxiv.org/pdf/math/0111245.pdf
The first result shows that deformations of standard complex tori are complex tori (i.e. $\mathbb C^n/\Gamma$ where $\Gamma\cong \mathbb Z^{2n}$). The second result shows that on $T^6$ there is an infinite dimensional family of complex structures.