Let \(K\) be a field complete with respect to some valuation 1.1 and let \(E=(E,\|\cdot \|)\) be a \(K\)-Banach space. \(K\)-Banach spaces \(E\) such that for each closed subspace \(D\) there exists a linear surjective projection \(P:E\to D\) satisfying \(\| Px\|\leq\| x\|\) for all \(x\in E\) are called norm Hilbert spaces (NHS). The authors introduce and characterize NHS in this paper. They also describe those NHS for which there exists a Hermitian form \((\cdot,\cdot)\) satisfying \(|(x,x)|= \| x\|^2\) for all \(x\). The main results proved are: Theorem 4.2: A norm Hilbert space is of countable type (and has an orthogonal base). Theorem 4.3: Let \(E\) be an infinite-dimensional \(K\)-Banach space with orthogonal base \(e_1,e_2,\dots\), let \(E= \bigoplus_{\sigma\in\Sigma} E_\sigma\) be its orthogonal decomposition. Then the following are equivalent: (i) \(E\) is a NHS. (ii) Each closed subspace has a closed complement. (iii) No subspace of \(E\) is linearly homeomorphic to \(c_0\). (iv) Each bounded subset of \(E\) is a compactoid. (v) Each \(E_\sigma\) is finite-dimensional. (vi) If \(\lambda_1,\lambda_2,\dots\in K\) are such that \(n\to\|\lambda_n e_n\|\) is bounded above then \(\lim_{n\to\infty} \lambda_n e_n= 0\). (vii) Each orthogonal sequence in \(E\) which is bounded above tends to \(0\). (viii) Each strictly decreasing orthogonal sequence in \(E\) tends to \(0\). Theorem 4.9: The dual of a NHS is an NHS and NHS are reflexive.