Let $\mathcal {N}$ be a non-Archimedean-ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD$^{k}$) functions and WLUD$^{\infty }$ functions at a point or on an open subset of $\mathcal {N}$. Then, we show under what conditions a WLUD$^{\infty }$ function at a point $x_0\in \mathcal {N}$ is analytic in an interval around $x_0$, that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD$^{k}$ and WLUD$^{\infty }$ to functions from $\mathcal {N}^{n}$ to $\mathcal {N}$, for any $n\in \mathbb {N}$. Then, we formulate conditions under which a WLUD$^{\infty }$ function at a point $\boldsymbol {x_0} \in \mathcal {N}^{n}$ is analytic at that point.

This work focuses on the ongoing research of lineability (the search for large linear structures within certain non-linear sets) in non-Archimedean frameworks. Among several other results, we show that there exist large linear structures inside each of the following sets: (i) functions with a fixed closed subset of continuity, (ii) all continuous functions that are not Darboux continuous (or vice versa), (iii) all functions whose Dieudonné integral does not behave as an antiderivative, and (iv) functions with finite range and having antiderivative.

We show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are weakly o-minimal. We also provide a short proof that algebraically closed valued fields with separated analytic structure (in any rank) are C-minimal.

We consider a meromorphic family of endomorphisms of degree at least 2 of a complex projective space that is parameterized by the unit disk. We prove that the measure of maximal entropy of these endomorphisms converges to the equilibrium measure of the associated non-Archimedean dynamical system when the system degenerates. The convergence holds in the hybrid space constructed by Berkovich and further studied by Boucksom and Jonsson. We also infer from our analysis an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of endomorphisms.

Let $\mathbf{k}$ be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let ${{D}_{1}},...,\,{{D}_{n}}$ be effective nef divisors intersecting transversally in an $n$-dimensional nonsingular projective variety $X$. We study the degeneracy of non-Archimedean analytic maps from $\mathbf{k}$ into $X\,\backslash \,\cup _{i=1}^{n}\,{{D}_{i}}$ under various geometric conditions. When $X$ is a rational ruled surface and ${{D}_{1}}$ and ${{D}_{2}}$ are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from $\mathbf{k}$ into $X\,\backslash \,{{D}_{1}}\,\cup \,{{D}_{2}}$. Using the dictionary between non-Archimedean Nevanlinna theory and Diophantine approximation that originated in earlier work with T. T. H. An, we also study arithmetic analogues of these problems, establishing results on integral points on these varieties over $\mathbb{Z}$ or the ring of integers of an imaginary quadratic field.

We prove field quantifier elimination for valued fields endowed with both an analytic structure that is $\unicode[STIX]{x1D70E}$-Henselian and an automorphism that is $\unicode[STIX]{x1D70E}$-Henselian. From this result we can deduce various Ax–Kochen–Eršov type results with respect to completeness and the independence property. The main example we are interested in is the field of Witt vectors on the algebraic closure of $\mathbb{F}_{p}$ endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first-order theory and that this theory does not have the independence property.

We prove a version of Montel’s theorem for analytic functions over a non-Archimedean complete valued field. We propose a definition of normal family in this context, and give applications of our results to the dynamics of non-Archimedean entire functions.

Let $K$ be the (real closed) field of Puiseux series in $t$ over ${\mathbf R}$ endowed with the natural linear order. Then the elements of the formal power series rings ${\mathbf R}[\![\xi_1,\dots,\xi_n]\!]$ converge $t$-adically on $[-t,t]^n$, and hence define functions $[-t,t]^n\to K$. Let ${\mathcal L}$ be the language of ordered fields, enriched with symbols for these functions. By Corollary 3.15, $K$ is o-minimal in ${\mathcal L}$. This result is obtained from a quantifier elimination theorem. The proofs use methods from non-Archimedean analysis.