Does oversampling improve processing gain?
You need to define what you mean by SNR and processing gain, because Marcus Muller and Qasim Chaudhari can both be considered correct depending on your definitions of SNR and processing gain. A processing gain is usually taken to be the SNR at the output of a system divided by the SNR at the input of a system. If you consider the SNR to be the signal power within the band of interest divided by the noise power within the band of interest, then there is no processing gain as you increase the sampling rate. This assumes an ideal (sinc) filter. In fact, without a perfect matched filter and perfect stopband attenuation (equal to zero), there is a processing loss caused by the out-of-band noise aliasing back into the spectrum after downsampling to the symbol rate. Although in all practical cases, this is very small and not really worth talking about. In other words, there is a fundamental limit to the SNR you can achieve by oversampling in this case. Without oversampling, it is likely that you will have poor filtering, and out-of-band noise may leak into the spectrum of interest (degrading it).

Now consider the SNR to be defined as the signal power divided by the total noise power contained in the samples (i.e. across the larger Nyquist bandwidth). In this case, you can reduce the noise level by filtering the excess noise that was added out-of-band. In other words, you will get a processing gain that is equal to the sampling rate divided by the true bandwidth of your signal (one over the baud interval for single carrier pulse shaped systems). This is often called a processing gain in ADC literature for example. Notice that if in your original samples, the matched filter design is good enough to reject out-of-band noise to negligible levels then there is no benefit in letting more noise into the system simply to reject it in order to obtain a high processing gain. It is probably worth noting that oversampling does result in a higher order matched filter which typically would result in a better stopband attenuation (making the excess noise less pronounced). This effect is very subtle and will never allow you to achieve a better SNR than you could by using a perfect matched filter (stopband attenuation = 0) at a lower sampling rate.