Shor's algorithm is often used as the argument. It can solve the factorization problem faster than any known algorithm for classical computers. Yet, we have no proof classical computers can't also factor integers efficiently.
Is there any actual proof quantum computers can solve some problems faster than classical computers?
Yes, Grover's algorithm shows you can use a quantum algorithm to find an element in an unordered database of size $N$ with high probability by querying the database only $O(\sqrt{N})$ times. Any classical solution that succeeds with high probability requires $\Omega (N)$ queries to the database.
It depends what you consider an actual proof, and what you mean by "faster". From a complexity theoretic perspective, the answer is no -- we don't have such a proof. BQP (the class of problems which can be solved efficiently by a quantum computer) is contained in PSPACE. Being able to prove a separation between BQP and PSPACE would also imply a separation between P and PSPACE, which is not known.
Note that Grover's algorithm only gives a square root speedup, so there is no contradiction.
you ask about "proof" which might be limited to a mathematical level, but the basic question goes much deeper than that. theoreticians will acknowledge its basically still an open question in general about the relative performance of quantum vs classical algorithms and there is probably no simple/ general answer, but with some expert consensus that Shors algorithm seems to be "unusually fast compared to expected best classical speed." fast factoring in a classical computer will break widely held cryptographic security assumptions such as the RSA system.
some of this is captured formally in the open complexity class question BPP =? BQP question. these are the analogous classical and quantum classes and the separation is unknown and an active area of research.
a closely related question is whether physically QM computers can be built that match the theoretical specifications and a few/ minority of scientists (aka "skeptics") are arguing that there may be noise or scaling laws that prevent QM scaling as envisioned in the theory. in a sense the ultimate "proof" of a QM computer speed must be a physical implementation. (this is similar to the way the Church-Turing thesis is theoretical but seems to ultimately tie into an assertion about physical implementations.) some researchers are talking about Church-Turing analogs in QM computing. see eg Church Turing thesis in a quantum world by Montanaro.
relevant to/ impinging on this question/ debate are ongoing substantial/ "heated" (scientific) attempts to benchmark the worlds current "largest" quantum computer by DWave. this is a big topic with a lot of related material, but for a relatively recent overview try D-Wave disputes benchmark study showing sluggish quantum computer / the Register
