Hölder and Lipschitz continuity of functions definable over Henselian rank one valued fields

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Consider a Henselian rank one valued field K of equicharacteristic zero with the three-sorted language L of Denef--Pas. Let f: A → K be a continuous L-definable (with parameters) function on a closed bounded subset A ⊂ Kⁿ. The main purpose is to prove that then f is Hölder continuous with some exponent s≥ 0 and constant c ≥ 0, a fortiori, f is uniformly continuous. Further, if f is locally Lipschitz continuous with a constant c, then f is (globally) Lipschitz continuous with possibly some larger constant d. Also stated are some problems concerning continuous and Lipschitz continuous functions definable over Henselian valued fields.