Does the Sobolev space $W^{1,p}(\Omega), p>2$ has a monotone basis?

Question

A Shauder basis in a Banach space is monotone if $\|P_{n}f\|\leq\|f\|,$ where $P_{n}$ is the projection to the sum of the first n elements of the basis. For Hilbert spaces this is always the case if one uses the orthonormal basis. Does the Sobolev space $W^{1,p}(\Omega),\, \infty>p>2$ has a monotone basis, or at least a basis that satisfy $\|P_{n}f\|_{W^{1,p}(\Omega)}\leq(1+O(1/n))\|f\|_{W^{1,p}(\Omega)}$? If not, how about if I assume that I am only interested in projecting subsets with higher regularity. For example, can I find a Shauder basis that satisfy $\|P_{n}f\|_{W^{1,p}(\Omega)}\leq(1+O(1/n))\|f\|_{W^{1,p}(\Omega)}$ for all $f$ with $\|f\|_{W^{1,\infty}(\Omega)}<C.$