Let $Q_i(x)$, $i\leq n$ be analytic $N\times N$ matrix-valued functions of $x\in D\subset\mathbb{R}^n$ a simply-connected domain, and let $X(x)$ satisfy $\frac{d}{dx_i}X(x)=Q_i(x)X(x)$. In the case $n=1$ it is easy to show that there is a unique analytic solution $X(x)$ using Picard iterates. Is the same true in multiple dimensions?
To be more specific, in one dimension the Picard iterates are $X_m(x)=\int_0^xQ_1(t)X_{m-1}(t)dt$. Will $X_m(x)=\sum_{i=1}^n\int_0^{x_i}Q_i(t)X_m(t)dt$ converge uniformly on compact subset of $D$ for the same reason?
Picard iteration
The iteration you describe does not really make sense. If $t$ is a vector variable, what does the integral $\int_0^{x_i}\dots\,dt$ mean? Perhaps you want to specify a polygonal path of integration, e.g., in two dimensions $$ X_m(x_1, x_2) =\int_0^{x_1} Q_1(t_1, 0)X_{m-1}(t_1, 0)\,dt_1 + \int_0^{x_2} Q_2(x_1, t_2)X_{m-1}(x_1, t_2)\,dt_2 $$ If this has a fixed point $X$, then indeed, $ \partial X/\partial x_2 = Q_2 X$, but the derivative with respect to $x_1$ is wrong. Whichever way you modify this, some of the partial derivatives will contain integrals differentiated with respect to a parameter, which does not fit the PDE.
No, better leave Picard iteration for ODE (which does include some evolution-type PDE that can be recast as ODE in a Banach space).
Existence issues
Consider the following example with $N=1$ and $n=2$: $$ \frac{\partial X}{\partial x_1} = 0;\quad \frac{\partial X}{\partial x_2} = x_1 X $$ (That is, $Q_1=0$ and $Q_2=x_1$). The solution must be independent of $x_1$, but then $x_2$-derivative can't depend on $x_1$. This forces $X\equiv 0$ which is of course always a solution, but this means we can't satisfy any nonzero boundary or initial conditions for this system.
This kind of example works for $N>1$, too. Things boil down to the PDE system being overdetermined: $N^2n$ scalar equations for $N^2$ unknown scalar functions.
Positive result
The Cauchy–Kovalevskaya theorem provides analytic solutions for similar systems, with a major difference: one equation per a component of unknown. In your setting, this means being able to satisfy $\partial X/\partial x_n = \dots$ (plus an initial condition).