Short answer
No. Although material may yield locally, it does not have to lead to failure. There is also a design with multiple pre-stressed walls, which may contain fluid at a higher pressure than material yield strength without any yielding.
Additional information
Elastic approach
Stresses in your answer are calculated based on fully elastic behaviour. When you increase the internal fluid pressure, the highest stress is the hoop on internal surface, but there is also radial stress component on the internal surface, which is basically minus fluid pressure. So if you apply Tresca yield criterion, elastic fluid pressure limit $P_{max,elastic,Tresca}$ for the cylindrical shell is just a half of its yield strength $R_e$ even for infinitely thick wall (you can see this from the following formula by taking internal diameter $D_i\rightarrow 0$).
$$P_{max,elastic,Tresca} = R_e\frac{D_e^2-D_i^2}{2D_e^2}$$
Although theoretically correct, I have never seen this formula in the practice. It is interesting, that formula for minimum thickness design based just on maximum hoop stress is used in practice, e.g., in EN 13480-3 for thick-walled pipes and possibly also in ASME standards (it was definitely used in the past, I am not sure about now).
Plastic limits
That being said, the vessel should not break when internal surface is yielding, because rest of the thickness is just fine. When you want to break it, you have to accomplish yielding in the whole thickness. Limiting pressure can theoretically go higher than yield strength of the material and the limiting factor would be allowable contraction of the material (I think this criterion can be found in ASME BPVC alternative rules):
$$P_{max,plastic,Tresca} = R_e\cdot \ln\left(\frac{D_e}{D_i}\right)$$
Since the second principal stress in cylindrical vessel is half of the first, the stress state is such that there is maximum difference between Tresca and Mises yield criteria. So the maximum pressure would be actually using Mises criterion (this one is used in EN 13480-3):
$$P_{max,plastic,Mises} = \frac{2}{\sqrt{3}}P_{max,plastic,Tresca}$$
If you need really high pressures, which would be difficult to achieve with single wall, you can use multiple walls which are carefully prestressed. Pressure limit of such a design is approaching plastic limit with number of walls going up. Note that plastic formula do not take into account hardening of the material, which may in some cases lead to significant increases of limit pressures.
Usual practice
In practice, most pressure vessel and piping design is done using formulae based on thin shell theory and maximum principal stress criterion, which leads to simple formula for limit pressure:
$$P_{max,simple} = 2R_e\frac{D_e-D_i}{D_e+D_i}$$
This lies somewhere between fully elastic and fully plastic states of the wall and is much easier to work with than other formulae. With slight modifications, it is used in many pressure vessel, boiler and piping standards.
Some additional remarks
Although formulae for thickness design are not based on elastic limit, there is a safety factor involved, so there may be no yielding. However, partial yielding can happen during hydrotest, where the test pressure is higher than design pressure, for example in EN 13445, safety factor may be 1.5, but then the test pressure may be 1.43 of design pressure. With sufficient material and fabrication quality, this is not a problem, on the contrary the vessel may actually be improved due to beneficial pre-stress and material hardening. This procedure is taken to its extreme in cryogenic pressure vessels (see slide 7), where the hydrotest does basically cold stretching and yield strength of special austenitic steels is increased significantly.
