The other answers to the effect that one needs big optics to see fine detail are indeed true for are true for conventional imaging optics that sense the electromagnetic farfield or radiative field i.e. that whose Fourier component at frequency $\omega$ can be represented as a linear superposition of plane waves with real-valued wave-vectors $(k_x,\,k_y,\,k_z)$ with $k_x^2+k_y^2+k_z^2 = k^2 = \omega^2/c^2$. This is the kind of field which the Abbe diffraction limit applies to and limits "eyes" like our own comprising imaging optics and retinas, or even compound eyes like those of an ant.
However, this is not the whole electromagnetic field: very near to the objects that interact with it, the electromagnetic field includes nearfield or evanenescent field components. These are generalised plane waves for which:
The component of the wavevector in some direction $k_\parallel$ is greater than the wavenumber $k$ and can thus encode spatial variations potentially much smaller than a wavelength;
The component of the wavevector $k_\perp$ orthogonal to this direction must therefore be imaginary, so that $k_\parallel^2 + k_\perp^2 = k^2$ can be fulfilled.
So such fields decay exponentially with distance from the disturbance to the electromagnetic field that begat them and thus cannot normally contribute to an image formed by an imaging system.
However, if you can bring your image sensors near enough to the disturbance, you can still register the detail encoded in the finer-than-wavelength evanescent components. This is the principle of the Scanning Nearfield Optical Microscope.
The near field optical microscope sensor can be extremely small indeed, so that a bacterium sized lifeform could register below-wavelength detail in the World around it with receptors built of a few molecules as long as the lifeform were near enough to the detail in question. Note that, when $k_\parallel > k$ that the fields decay like $exp(-\sqrt{k_\parallel^2-k^2} z)$ with rising distance $z$ from their sources. So there is a tradeoff between how much finer than a wavelength we can see with such a sensor, and how near to the source we need to be to see it. If we want to see features one tenth of the wavelength of seeable light, then $k\approx 12{\rm \mu m^{-1}}$ and $k_\parallel \approx 120{\rm \mu m^{-1}}$, so that the amplitude of the nearfield decays by a factor of $e$ for each hundredth of a wavelength distant from the source the detector is. Thus we lose about 10dB signal to noise ratio for every hundredth of a wavelength distance that separates the detector and source. So to sense such fine detail (50nm structures) from a micron away would need extremely strong light sources, so that the detectors would have a very clean signal.
Of course, the above is an extreme example, but if you're a bacterium sized lifeform directly sensing the field using a finely spaced array of molecular sensors, you may well be able to "see" below-wavelength features of the World in your immediate neighbourhood. Moreover, it is possible to conceive of a tiny creature "feeling" its neighbourhood using molecular atomic force microscopes.
So, yes, if you include all physics and heed the proviso that you must get up really close to the sensed objects, it would be possible for a bacterium sized lifeform to see below-wavelength detail in its immediate neighbourhood, maybe even individual atoms if we include atomic force sensing.
Of course, packing all the signal processing "brain" into the lifeform needed to understand this information might be another matter altogether.
