Oscillations in a first-order discrete-time linear systems
For continuous-time systems, a pole at location $s_0=\sigma_0+j\omega_0$ will create a time-domain contribution of the form

$$e^{s_0t}=e^{\sigma_0t}e^{j\omega_0t}\tag{1}$$

which is a damped oscillation if the pole is in the left half-plane (i.e., $\sigma_0&lt;0$), and if the pole is not on the real axis (i.e., $\omega_0\neq 0)$. For $\omega_0=0$ there is no oscillation. This is the case for a real-valued first-order system, which can only have a real-valued pole.

For discrete-time systems, a pole at $z_0=re^{j\omega_0}$ ($r\ge 0$) causes a sequence of the form

$$z_0^n=r^ne^{j\omega_0n}\tag{2}$$

If the pole is inside the unit circle (i.e., $r&lt;1$) that sequence will decay in amplitude. Note that the sequence will not oscillate only if $\omega_0=0$, i.e., if the pole is on the positive real axis. If the pole is real, but if it is negative, we have $\omega_0=\pi+2\pi k$, i.e., we have an oscillation with maximum frequency, corresponding to an alternating sequence.

In sum, for continuous-time systems if the pole is on the real axis, it will not cause oscillations. For discrete-time systems, there are no oscillations only if the pole is on the positive real axis. A pole on the negative real axis causes oscillations with maximum frequency. Consequently, a discrete-time real-valued first-order system can exhibit oscillations if the pole happens to be on the negative real axis.

For additional insight consider the exact mapping between the $s$-plane and the $z$-plane:

$$z_0=e^{s_0T}\tag{3}$$

where $T$ is the sampling interval. Inversion of $(3)$ gives

$$s_0=\frac{\ln(z_0)}{T}\tag{4}$$

A $z$-plane pole $z_0=-r$ ($r&gt;0$) on the negative real axis maps to a complex valued pole in the $s$-plane:

$$s_0=\frac{\ln(-r)}{T}=\frac{\ln(r)}{T}\pm j\frac{\pi}{T}\tag{5}$$

The frequency $\omega_0=\pi/T$ corresponds to half the sampling frequency, i.e., the maximum frequency of the corresponding discrete-time system.

As a final note, oscillations have nothing to do with whether the system is stable or not (stability in the bounded-input bounded-output sense). For stability, it only matters if the contributions of the poles to the output signal decay or not.