Uniqueness and Value Sharing of Differential Polynomials over a Non-Archimedean Field

Let $$\mathbb K$$ be an algebraically closed field of characteristic zero, complete for a nontrivial non-Archimedean absolute value. Let $$a,b\in\mathbb K\setminus\{0\}$$ and let $$n,k,p$$ be positive integers. We study non-constant functions $$f$$ and $$g$$ whose differential polynomials $$f^n+a(f^{(k)})^p$$ and $$g^n+a(g^{(k)})^p$$ share the value $$b$$ $$\mathrm{CM}$$ . For entire functions, if $$n\geq p+2$$ and $$\gcd(n,p)=1$$ , then $$f=g$$ unless $$f$$ and $$g$$ are proportional polynomials of degree at most $$k$$ . For meromorphic functions, if $$n\geq 2p(k+1)+8$$ , then $$f$$ and $$g$$ satisfy one of three explicit algebraic relations. These results extend earlier work of Tuan and Quang and, for $$p=1$$ , improve their bounds on $$n$$ .