In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?

This is cross-posted and answered on MO here.

Let $(X,d)$ be a metric space. Say that $x_nin X$ is a P-sequence if $lim_{nrightarrowinfty}d(x_n,y)$ converges for every $yin X.$ Say that $(X,d)$ is P-complete if every P-sequence converges. Problem 1133 of the College Mathematics Journal (proposed by Kirk Madsen, solved by Eugene Herman) asks you to prove that $$text{compact}Longrightarrowtext{P-complete}Longrightarrowtext{complete}$$ and that none of these implications go both ways. The implications follow by showing that $$text{sequence}Longleftarrowtext{P-sequence}Longleftarrowtext{Cauchy sequence},$$ since a P-sequence (and thus a Cauchy sequence) converges iff it has a convergent subsequence. To give counterexamples to the converses, there are several possible directions. My question specifically involves normed vector spaces (although it is overkill for the original problem).

For any $ngeq 0$, any norm on $mathbb R^n$ induces a P-complete metric. This distinguishes compactness and P-completeness, since $mathbb R^n$ obviously isn’t compact when $n>0$. To differentiate P-completeness and completeness, we can note that a Hilbert space is P-complete iff it is finite-dimensional (otherwise, we take a non-repeating sequence of vectors from an orthonormal basis and get a P-sequence that doesn’t converge). I wonder if other infinite-dimensional normed spaces (necessarily Banach) might be P-complete. But my knowledge of Banach spaces is very limited, so I don’t have much intuition about what examples to try. Also, the property of P-completeness (unlike compactness and completeness) is not closed-hereditary, so we can’t just try an something by embedding it in a larger example.

Question: What is an example of an infinite dimensional, P-complete Banach space?

Examples I tried:

• $ell^p$ spaces for all $1leq p<infty$. They are not P-complete, since the sequence $e_n=(0,dots,0,1,0,dots)$ is a P-sequence but not Cauchy.
• $C(X)$ for $X$ compact Hausdorff, first-countable and infinite. There must be an accumulation point $pin X$. We can take a sequence of bump functions $f_k$ converging (pointwise) to the characteristic function $chi_p$. For any $gin C(X)$, we have $lim d(g,f_k)=||g-chi_p{||}_infty$. Thus $(f_k)$ is a P-sequence that does not converge (uniformly), because the pointwise limit is discontinuous.