Non Periodic, Deterministic Power Signals
Consider the signal
$$x(t) = \sum_{n=-\infty}^\infty \operatorname{rect}(t-n)\sin(2\pi nt)$$ which consists of $|n|$ periods of the signal $\sin(2\pi nt)$ in the interval $\left(n-\frac 12,n+\frac 12\right)$ for each integer $n$. Clearly, $x(t)$ is not periodic. Equally clearly, $x(t)$ is a power signal with average power $\frac 12$, and thus it is a nonperiodic but deterministic power signal. I leave it to the OP to determine whether the signal in question has a Fourier transform in the generalized sense (meaning it involves impulses) or not.