The unique crystalline protection of the surface states in topological crystalline insulators has led to a series of predictions of strain-generated phenomena, from the appearance of pseudo-magnetic fields and helical flat bands to the tunability of Dirac surface states by strain that may be used to construct 'straintronic' nanoswitches. However, the practical realization of this exotic phenomenology via strain engineering is experimentally challenging and is yet to be achieved. Here, we have designed an experiment to not only generate and measure strain locally, but also to directly measure the resulting effects on Dirac surface states. We grew heteroepitaxial thin films of topological crystalline insulator SnTe in situ and measured them using high-resolution scanning tunnelling microscopy to determine picoscale changes in the atomic positions, which reveal regions of both tensile and compressive strain. Simultaneous Fourier-transform scanning tunnelling spectroscopy was then used to determine the effects of strain on the Dirac electrons. We find that strain continuously tunes the momentum space position of the Dirac points, consistent with theoretical predictions. Our work demonstrates the fundamental mechanism necessary for using topological crystalline insulators in strain-based applications.